Real-Time Probabilistic Collision Avoidance for Autonomous
Vehicles, Using Order Reductive Conflict Metrics
by
Thomas Jones
M.Sc. University of Stellenbosch (1999)
B.Eng. University of Stellenbosch (1997)
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
MASSACHUSETTS INSTITUTE
Philosophy
of to
Doctor D
OF TECHNOLOGY
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at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2003
Massachusetts Institute of Technology 2003. All rights reser
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LIBRARIES
Author
Department of Aeronautics and Astronautics
I
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June. 9003
Certified by .............................
i John J Deyst
Professor of Aeronautics and Astronautics
Certified by.................
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...............................
James K. Kuchar
Associate Professor of Aeronautics and Astronautics
ThesjjCommittee Member
Certified by.........
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Brent Appleby
Principal Member of the Technical Staff, Draper Laboratory
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ilhesis Coirnttet
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Certified by ........
Fesq
Lorraine
JPL
Principal Engineer, NA
Thesis Cnujnittee Member
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Emilio Frazzoli
Assistant Professor of Aeronautics and Astronautics, UIUC
Thesis Committee Member
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Certified by ............................
A ccepted by ....................
.......................
Edward M. Greitzer
H.N. Slater Professor of Aeronautics and Astronautics
Chairman, Committee on Graduate Students
AERO
4
Real-Time Probabilistic Collision Avoidance for Autonomous Vehicles,
Using Order Reductive Conflict Metrics
by
Thomas Jones
Submitted to the Department of Aeronautics and Astronautics
on June, 2003, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
Contemporary collision avoidance systems such as the Traffic Alert and Collision Avoidance
System (TCAS) have proven their effectiveness in the Commercial Aviation (CA) industry
within the last decade. Yet, TCAS and many systems like it represent attempts at collision avoidance that do not fully recognize the uncertain nature of a conflict event. Most
systems circumvent probabilistic representation through simplifying approximations and
pre-compiled notions of hazard space, since probabilistic representation of collision in three
dimensions is considered to be an intractable problem.
Recent developments by Kuchar and Yang[70] and Paielli and Erzberger[50] have shown
that collision avoidance may be cast as a probabilistic state-space problem. Innovative solution approaches may then allow systems of this nature to probe collision risk in real-time,
based on real-time state estimates. The research documented in this thesis further develops the probabilistic approach for the non-cooperative, two-vehicle problem as applied in
real-time to autonomous aircraft. The research is kept in a general form, thereby warranting application to a wide variety of multi-dimensional collision avoidance applications and
scenario geometries.
The work primarily improves the state of the art through the creation of order reductive
collision metrics in order to simplify the intractable problem of multi-dimensional collision
risk calculation. As a result, a tractable, real-time, probabilistic algorithm is developed for
the calculation of collision risk as a function of time.
The collision avoidance problem is contextualized not only within the realm of recent
research within the CA industry, but is also likened to such concepts as the first passage
time problem encountered in physics, and the field of reliability theory often encountered in
civil and mechanical engineering problems. Yang's method of solution, a piece-wise straightline Monte-Carlo approach to state propagation, is extended with a model-predictive, finite
horizon risk accumulation algorithm. Through this extension we are capable of modelling
collision risk for linear(-ized), time-variant, dynamic vehicle models and control strategies.
A strategy is developed whereby the advantage of delayed collision avoidance action is
calculated and it is framed as an extension of the notion of system operating characteristics (SOCs). The complexity of the probabilistic representation is reduced by application
2
of quadratic conflict metrics. The numerical complexity can be reduced from O(N ") to
3
O(Nlog 2 (N)) at each time step within a finite horizon time interval.
Risk calculation errors due to numerical and stochastic approximations are quantified.
An applicability test is also devised whereby a vehicle's dynamic model and control characteristics may be used to calculate risk error estimates before implementing the bulk of the
algorithmic solution. Various other applications of the work, outside the scope of collision
avoidance, are also identified.
Thesis Supervisor: John J. Deyst
Title: Professor of Aeronautics and Astronautics
4
Acknowledgments
This thesis is not the culmination of my work at MIT, but rather a testimony to the support, good nature and spirit of a number of incredible people without whom I could not
have gotten by.
Let me first thank the MDPP group with whom I have spent the past 4 years. They
are my family away from home. Sanghyuk "San-Seng" Park (Anyoong Hase-Yo!), Francois
"Ou Perd" Urbain (Los Tabernakos!), Sarah Saleh, Damien Jourdan, Damian Toohey, Jason Kepler, Richard Poutrel, Anand Srinivas and Alexander Omelchenko. Special thanks
to Sanghyuk and Francois, for their wisdom, friendship and kindred spirits.
Then there is my thesis committee, each of whom have played a valuable role in my
education at MIT: Prof. John Deyst, Dr. Brent Appleby, Dr. Lorraine Fesq, Prof. James
Kuchar and Prof. Emilio Frazzoli. Special thanks to Lorraine, who arrived at MIT at the
same time as I did, for your open door, friendship and the skills you have taught me. Also a
very special thanks to Prof. Deyst, my mentor, who took many chances on me and provided
me with so many opportunities, again and again. I trust that I have not disappointed you.
There are also a number of silent partners who surfaced at the most critical times to provide support and guidance. I wish to thank Prof. Danielle Veneziano, for sound advice and
his willingness to help; Dr. Lee Yang, for proof reading the thesis and his valuable inputs;
Sean George, for his assistance in my day to day project research; Donald ("The Don")
Wiener, for his kindness, wisdom and support; Dick Perdichizzi, for his constant and easily
offered advice; and Phyllis Collymore, for sorting out anything that could possibly go wrong.
I need to thank a number of individuals who believed in me enough to support my coming to MIT in the first place. Prof. Jan du Plessis, dankie vir sewe jaar van ondersteuning,
aansporing, vertroue en advies; Dr. Jan Roodt, for his wisdom and support; and the NRF,
in the personage of Rose Robertson, for the financial support which sustained me when I
needed it the most.
5
Then there are my friends in Boston: Ernst Scholtz, Amanda Hollenbaugh, Tom Reynolds,
Halyley Davison, Blake and Jeanine Booysen, Simon Nolet (Je me souviens Quebecois, mon
ami!), Xueen Yang and Lodewyk Steyn, sources of constant support and unforgettable times.
Also my friends back home, Japie and Madri Engelbrecht, Nelius Goosen, Gerhard le Roux
and P.C. Visagie, for their warm friendship and for their roles in motivating me to study at
MIT.
I wish to thank my mom and dad for believing in me and for having brought me up to
be the man I am today. Also many thanks to my new family, Lex and Andy de Vries, soon
to be called mom and dad too.
Then there is Lisa Ann de Vries, or simply Leesie. I thought I'd be going to the States
for a Ph.D, but I also found you. Thank you for your love and kindness, constant support
and for all those adventures past and those yet to share. I love you dearly and this thesis
is dedicated to you. Leesie, I'm coming home now!
6
Contents
1
2
1.1
Background . . . . . . . . . . . ..
. . . . . . . . . . . . . . . . . . . . . . .
17
1.2
Uniqueness of the Autonomous UAV Problem . . . . . . . . . . . . . . . . .
19
1.3
Research Objectives
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
1.4
Thesis Overview
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
1.5
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
25
Alerting System Concepts
2.1
General Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2
Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.1
State Space Representation . . . . . . . . . . . . . . . . . .
28
2.2.2
Standard Definitions and Framework . . . . . . . . . . . . .
30
2.2.3
Methods of Approach
. . . . . . . . . . . . . . . . . . . . .
31
A Probabilistic Approach . . . . . . . . . . . . . . . . . . . . . . .
34
2.3.1
Concerning Alert Space . . . . . . . . . . . . . . . . . . . .
34
2.3.2
Similarity of Collision Avoidance to Alternative Problems .
36
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
2.3
2.4
3
17
Introduction and Background
Expanding the Probabilistic State Space Approach
3.1
3.2
43
. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .
44
3.1.1
State-Space Formulation . . . . . . . . . . . . . . . . . . . .
44
3.1.2
Propagation of State Distributions . . . . . . . . . . . . . .
46
Propagating and Accumulating Risk . . . . . . . . . . . . . . . . .
48
3.2.1
The Single-Vehicle-Single-Obstacle Problem . . . . . . . . .
48
3.2.2
Accumulation of Risk
. . . . . . . . . . . . . . . . . . . . .
49
Estim ation
7
3.2.3
3.3
3.4
Propagation Procedure
. . . . . . . . . .
53
Time-delayed Avoidance . . . . . . . . . . . . . .
55
3.3.1
System Operating Characteristic . . . . .
55
3.3.2
Condensed SOC . . . . . . . . . . . . . .
57
Chapter Summary . . . . . . . . . . . . . . . . .
61
4 Tractable Risk Propagation Through Quadratic Collision Metrics
4.1
4.2
4.3
4.4
5
. . . . . . . . . . . . . . . . . . . . . . . .
63
4.1.1
Simplifying Assumptions . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.1.2
Mathematical Reduction . . . . . . . . . . . . . . . . . . . . . . . . .
65
4.1.3
Simplifying Approximations . . . . . . . . . . . . . . . . . . . . . . .
66
Induced Error on Probability Calculations . . . . . . . . . . . . . . . . . . .
77
4.2.1
Approximation Error . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
4.2.2
Numerical Approximation . . . . . . . . . . . . . . . . . . . . . . . .
89
4.2.3
Quadratic Approximation of Collision Bounds . . . . . . . . . . . . .
91
4.2.4
Time Discretization Error . . . . . . . . . . . . . . . . . . . . . . . .
92
Time and Resolution Issues . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.3.1
Finite Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4.3.2
Avoidance Trajectory Discretization Step Size . . . . . . . . . . . . .
99
4.3.3
Special forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101
Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
Application Examples
105
5.1
PCUAV NGM-II and a Stationary Hazard . . . . . . . . . . . . . . . . . . .
106
5.1.1
Vehicle and Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . .
106
5.1.2
Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
Two Large Commercial Transports . . . . . . . . . . . . . . . . . . . . . . .
113
5.2.1
Vehicle and Scenarios
. . . . . . . . . . . . . . . . . . . . . . . . . .
113
5.2.2
R esults
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
5.2
5.3
6
Order Reduction of Convolution
63
Chapter Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
Conclusions and Future Work
121
6.1
121
Primary Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
6.2
Applicability and Comparisons . . . . . . . . . . . . . . . . . . . . . . . . .
122
6.3
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123
6.4
Future Work ........
127
...................................
6.4.1
General State Avoidance . . . . . . . . . . . . . . . . . . . . . . . . .
127
6.4.2
Control and Trajectory Scheduling for Minimal Risk . . . . . . . . .
127
6.4.3
n-Multiple and Compound Vehicles . . . . . . . . . . . . . . . . . . .
128
139
A Matlab Simulation Code
A.1
Sample Code for Calculation of Probability of Collision Within One Finite
Horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2
139
Sample Code for Calculation of Statistics of Two Dependent Quadratic Forms,
"findrho.m ": . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
146
10
List of Figures
1-1
Future State Uncertainty as a Result of Uncertain Intent
. . . . . . . . . .
20
2-1
Block Diagram of the Conflict Detection and Resolution Process . . . . . .
26
2-2
A 2-Dimensional State Space Representation of the Avoidance Problem
. .
29
2-3
Flowchart of Different Alerting Outcomes . . . . . . . . . . . . . . . . . . .
30
2-4
Possible Different Outcomes of an Alert in a State-Space Representation
32
2-5
Flowchart Representation of Successful Avoidance and Unnecessary Alerts .
35
2-6
An Illustration of the Probabilistic Alert Space . . . . . . . . . . . . . . . .
37
2-7
A 2-Dimensional Representation of Hazard Space in Reliability Theory . . .
38
2-8
An Illustration of the First Passage Time Problem for Two Absorbing Bounds 40
3-1
A Basic 2-Dimensional Representation of the Risk Accumulation Process,
.
Illustrated With 1-Dimensional Triangular State Distributions, for Simplicty
50
3-2
Progression of fx, (x) IB[o,k_1] Over One Time Step from t = to, in 2 Dimensions 51
3-3
5-Step fX(tk+1)(x)IB[o,k] Propagation and Calculation of Pc(k) (1-Dimensional) 54
3-4
Different Shapes of SOC Curves . . . . . . . . . . . . . . . . . . . . . . . . .
56
3-5
An Example Showing the Benefit of Delayed Avoidance Action . . . . . . .
58
3-6
The Unpredictable Nature of the SOC Curve for t > to . . . . . . . . . . . .
59
3-7
Creation of Compound Avoidance Maneuvers . . . . . . . . . . . . . . . . .
60
4-1
Region in the 7k-Attk Plane where yk+1
yk + Atik, Used to Describe
. . . . . . . . . . . . . .
66
4-2
4-Step fr(tk+l)(7)|B[kk Propagation and Calculation of Pe(k) . . . . . . . .
70
4-3
Relationships Between Ptre(t), Peb
. . . . . . . . .
79
4-4
Calculating Pjb(t) Over a Finite Time Horizon Interval . . . . . . . . . . . .
80
4-5
Contour Plot of fr
(y, ') with Correlation Line . . . . . . . . . . . . . . .
82
Frk+1(7)
. . . . . . .. . .
. .. .
.
11
- - - - -.
perr(t) and Pb(t)
Rak vs. y for the Simulation of a Vehicle Flying Towards the Origin
. . . .
88
4-7 Factors Influencing the Size of t, . . . . . . . . . . . . . . . . . . . . . . . .
97
4-8
2D Example for Finding the Minimum Safe Propagation Time Window
. .
99
5-1
The Two PCUAV NGM-II Application Example Approach Scenarios . . . .
108
5-2
Risk Accumulation for PCUAV NGM-II Hazard Fly-by
. . . . . . . . . . .
110
5-3
Risk Accumulation for PCUAV NGM-II Head-on Hazard Approach . . . . .
113
4-6
5-4 The Two Transport vs. Transport Application Example Approach Scenarios 116
5-5
Risk Accumulation for Transport Fly-by . . . . . . . . . . . . . . . . . . . .
117
5-6
Risk Accumulation for Transport Head-on Approach . . . . . . . . . . . . .
118
12
List of Abbreviations
ATM
Air Traffic Management
CA
Commercial Aviation
CD
Correct Detection
CR
Correct Rejection
CSOC
Condensed System Operating Characteristic
DMOD
Distance Modification (see TCAS)
FA
False Alarm
FAA
Federal Aviation Authority
FFT
Fast Fourier Transformation
FMC
Flight Management Computer
FI
Fourier Inversion
FOSM
First Order, Second Moment; As superscript indicates corrected to FOSM
IDFT
Inverse Discrete Fourier Transformation
IFFT
Inverse Fast Fourier Transform
IID
Independent, Identically Distributed
lb
Lower Bound; As superscript indicates upper bound value
LTI
Linear, Time Invariant
MD
Missed Detection
MDPP
MIT/Draper Technology Development Partnership Program
MPT
Mean Hazard Passage Time
NGM-II
New Generation Mini Vehicle Two (a PCUAV Aircraft)
PCUAV
Parent-Child Unmanned Aerial Vehicle Project
PDF
Probability Density Function
SA
Successful Avoidance
SOC
System Operating Characteristic
TCAS
Traffic Alert and Collision Avoidance System
UA
Unnecessary Avoidance
UAV
Un-crewed Aerial Vehicle
ub
Upper Bound; As superscript indicates upper bound value
UCAV
Unmanned Combat Aerial Vehicle
13
14
List of Nomenclature
Symbol
Meaning
At
Time discretization step size
J(t -
Dirac delta function occurring at t
AEak]
Change in expected value of a because of truncation after t =tk
Ay, A', AF, At
Equivalent to At -7, At -
-y
7 = xTBx quadratic collision metric
70
Boundary of Df such that 7 < 70 constitutes collision
Or,(-r)
Characteristic function of 7
Qak
FOSM error in calculation of a at t =
Ai
Avoidance maneuver delayed by
Da(t), Df(t), D8 (t)
Domains of alerting, failure and safety respectively
DCT
i'th Discrete Time Compound (Avoidance) Trajectory
E[a]
Expected value of some random variable a
fx(t) (x)
PDF of x at time t
fx(ta)(x)|B[b,c)
PDF of x at time ta given no collision within [tb, tc]
fr(ta) (7) IB[b,c]
PDF of 7 at time ta given no collision within
fx~a(x)
fx(t,)(x)|B[o,k-1i, zeroed where x E Df, and Bayes normalized
fB()
fr(tk)(7)B[k-1,k-1],
[1
j,
=
T
At - F, At -
respectively
tk
i.tAD
[tb,
te]
zeroed where 7 < 70
- u(x - Xk)]fx(tk)(x)|B[o,k-1]
and u(x
- Xk)fx(tk)(X)|B[o,k-1|
respectively
9Ak(7), 9Bk (7)
[1 - u(7 - 7k)]fr(tk)(7)IB[k-1,k-1] and u(7 - 7k)fr(t)(7)IB[k-1,k-1]
respectively
K
Number of discrete simulation time steps
max[t.,t [f(t)]
Maximum value of f(t) between ta and
I
Number of Discrete Time Compound Trajectories
th/tAD + 1
tb, ta < tb
N(m, E)
Normal distribution with mean m and covariance E
P(A)
Probability of occurrence of event A
Pc(ta)
Total probability of collision accumulated over the interval [to, ta]
Pc(k)
Pc(k)
Pc(tk)
15
PC'(k), Pc'(k), Petrue (k)
Lower bound, Upper Bound and True probabilities of collision
respectively, accumulated over the interval [to,
PCmax
tk]
Maximum acceptable safe probability of collision for a vehicle
pA, pN
Pc(th) given that avoidance action is taken or not, respectively
R"
n-dimensional Real space
T1, T2
P(SA) and P(UA) Alerting Thresholds respectively
t
Time
t, th
Simulation start time and finite horizon (end)time respectively
tc
Maximum simulation time length, bound by real-time tractability
td
Unforeseen intent change implementation delay interval
tk
Time at k'th discrete time step
tw
Finite horizon simulation time window length
u(x - Xa)
Unit step function at x = x,
v(t)
White measurement noise
w(t)
White process noise
xi(t)
i'th state variable
x(t)
State vector, dim(x(t)) = n
XOL(t)
Open loop state vector
16
Chapter 1
Introduction and Background
1.1
Background
More than a century ago we saw the dawn of an era of crewed flight. We have now embarked
upon a new era, where the wonders of aviation technology have become commonplace and
technological advances, especially in electronics and software, have changed the face of the
aviation industry over the last twenty years[38].
The 1980's and 90's saw the introduction of so-called glass cockpits and later flight directors and traffic alert systems, increasingly pushing towards automation in the commercial
aviation (CA) cockpit. Even though the human flight crew does play a critical role, few
passengers know that autopilots and flight management computers (FMCs) are responsible
for flying them most of the way between airport destinations. Recently, various issues such
as loss of situational awareness have been under scrutiny and have been linked to human
pilots' automated cockpit systems[36]. This opens the door for automated collision avoidance and air traffic management (ATM) systems that would be capable of playing the role
of not only an alerting system, but also would automatically solve inter-vehicle conflict.
Military aviation has also undergone a dramatic transformation during the end of the
20th century. More and more Un-crewed Aerial Vehicles (UAVs)[47] are being developed
and many of those such as Predator and Global Hawk have become household names. The
state of current technology and a political climate, where risk of the loss of human life
is of increasing concern, are fuelling the inevitable growth of the UAV industry. Some
17
of the advances so far include un-crewed combat vehicles (UCAVs), un-crewed reconnaissance vehicles and multi-role UAVs like Predator[47] [7] [1] [13] [46]. Recent developments at
MIT, such as the Parent-Child Un-crewed Aerial Vehicle (PCUAV) [63] [51] [37] system have
shown that multi-tier co-operative UAV systems are prime candidates for the collection of
up-close reconnaissance information. This is especially true when dealing with cluttered
and inhospitable terrain such as caves and forests where high-altitude surveillance proves
to be of limited use. These missions were always considered to be too risky for piloted flight.
Many UAVs are still piloted by human operators, but the added separation between
the pilot and the vehicle and limited sensory perception (esp. visual) only increases the
frequency and severity of loss of situational awareness. Add to this the expense of having
highly skilled operators for each independent vehicle and large vehicle support teams and
we see that the opportunities for added autonomy abound. As vehicles become increasingly
self-sufficient, one such opportunity arises for the creation of automated collision avoidance
systems for autonomous vehicles.
Most collision avoidance systems are developed for the CA industry, in an attempt to
boost safety while relaxing the in-trail spacing currently observed to maintain adequate
traffic separation, usually during piloted flight[67] [56]. There is a need to create automated
collision avoidance systems, especially for autonomous vehicles, that are on par with the
most recent developments in alerting systems for the CA sector[43] [70][69].
At present, UAVs and especially autonomous UAVs are required to remain outside of
FAA controlled airspace. Except in very special circumstances, air traffic control authorities
do not allow autonomous, or even un-crewed, aircraft general access to such airspace, since
they are considered to pose a threat to civilian traffic. Vehicles would need not only trusted
control systems and trajectory scheduling capability, in order for this ban to be lifted, but
would also require the ability to adapt to unforseen hazardous situations. Automated collision avoidance therefore brings us one step closer towards opening a door to commercial
airspace for autonomous UAVs. Until then, the UAV industry will be constrained to specialized missions, especially of a military nature.
18
There are a number of reasons why UAVs provide a remarkable testing ground for
automated collision avoidance systems. Foremost is the lack of human presence aboard
a UAV, thereby allowing us to attempt a large number of solutions without putting life
in danger. The resulting development rate of systems that integrate onto UAVs and the
reduced certification requirements make this platform an excellent breeding ground for the
development of new avionics systems. Also, there is an increasing need for the interaction
between UAVs and crewed vehicles, a prospect that can naturally make any pilot nervous.
Automated collision avoidance systems may make such interaction feasible in the near
future. Finally, there is the issue of using UAVs as a proving ground for avoidance systems
that might eventually make their way into a crewed cockpit.
1.2
Uniqueness of the Autonomous UAV Problem
Inter-vehicle collision avoidance systems are naturally designed to be most useful under
conditions of increased air traffic density, where there is an associated increase in the risk
of conflict. The traditional school of thought dictates that avoidance systems be limited to
acting primarily as alerting systems, since pilots are especially vigilant or fully in control of
an aircraft under conditions of increased traffic. Pilot response then dictates the measure
of conflict resolution. This view still holds for the CA industry, and rightly so. Humans
can be skilled operators and may evaluate measures of safety and apply measures of evasion
and control that are far beyond the capabilities of traditional automated systems. A more
probabilistic view suggests that pilots also add a measure of decision-making capability
to the collision avoidance problem that is at least not fully correlated with the existing
alerting system. This may improve the avoidance system's performance. We have grown
accustomed to their proven safety record and thus airlines, passengers and authorities feel
more comfortable when a human supervises and/or controls a vehicle.
It is partly the complexity of the nature of a human operator that also hampers the
development of adequate collision avoidance systems. There is a very definite uncertainty
associated with the intent of a human operator. Collision avoidance by definition includes
the capability to project future vehicle state in order to be able to predict future inter-vehicle
conflict. It is the uncertainty of intent that hampers the accuracy of conflict detection and
19
~
-~A
. ID= -i.
-
most alerting systems need to be designed around this problem [41] [42] [69] [70].
Autonomous UAVs provide a unique opportunity to create conflict detection systems
where more certainty exists about vehicle intent. Most auto-pilots are deterministic and
models of their operation may be created with an accuracy far exceeding that of human
operator models. There remains a measure of uncertainty though and its effects on the
avoidance system are discussed in Section 4.3.1. Figure 1-1 illustrates the typical conceptual difference in future state prediction uncertainty between manually and autonomously
piloted vehicles in the presence of a hazardous area. Such a hazardous area is referred to
as a domain of failure Df (t).
Future State
Uncertainty:
With Autopilot
Without
Autopilot
Vehicle State
Uncertainty at t =to
Figure 1-1: Future State Uncertainty as a Result of Uncertain Intent
Another troublesome aspect of alerting systems is that there is an interaction between
the performance of collision avoidance and an alerting system's tendency to issue false
alarms. Section 2.1 provides more detail about this concept which was also analyzed by
Kuchar[41]. In short, false alarms lead to mistrust of a system by human operators. It is
this mistrust of a system that reduces its effectiveness, in accordance with such principles as
risk homogenization and negative reinforcement[8]. Pilots tend to become less inclined to
20
_
.-
- -
take action after false alerts have been issued, thereby increasing the risk of collision when
an alert turns out to be valid.
Autonomous UAVs do not suffer from the same adverse consequences of interaction
between false alarm rates and system performance. In a well-segmented system, one subsystem, namely the conflict alerting software, would decide whether a collision is imminent
and alert when a threshold is exceeded. The vehicle's guidance and control sub-system
would then implement an avoidance trajectory suggested by the alerting system. Even
though false alarm rates may increase the number of trajectory changes and the associated
trajectory cost, the autonomous control system would not normally distrust the validity
of alerts, since it does not usually have this capability. After a flight we may analyze the
performance of an alerting system and adjust it as best possible, but in-flight alerts are
always heeded. The result is that we may design a system where the cost associated with
false alarms is much smaller than that for piloted platforms.
1.3
Research Objectives
This thesis details the development of a real-time, automated collision avoidance system
for autonomous vehicles. We focus on a non-cooperative two-vehicle conflict scenario. First
we will show how this problem is different from conflict resolution and alerting systems
usually developed for crewed flight. It will be shown that these differences may be exploited
at a basic level of system design and that a new breed of collision avoidance systems may
be developed that is based on the creation of collision risk estimates within a finite time
horizon interval.
The work is primarily developed from the probabilistic framework and principles laid
down by Kuchar and Yang [42] [70) [41] [69]. The initial development also draws on methods
described by Paielli and Erzberger[49][50]. An emphasis is placed on progressing from basic
probability theory and we show that the solution is computationally intractable. It is, after
all, this intractability that usually leads researchers to apply ad-hoc methods of approximation and probabilistic solutions such as the Monte-Carlo method. Instead, we show how the
complexity of the solution space may be reduced considerably through the application of
21
order reductive collision metrics, such as describing collision hazards as multi-dimensional
quadratic forms. We then apply well-founded approximations to the remaining problem in
order to develop a tractable probabilistic solution to real-time collision avoidance.
It is also shown how the System Operating Characteristic (SOC) representation developed by Kuchar [42][41] may be expanded to evaluate the non-trivial probabilistic effects
of delaying collision avoidance action.
Two example applications are simulated, one for a small PCUAV vehicle and one for
large commercial transports. A number of future avenues of research are discussed, including
application to automated landing systems and other automated maneuvers where there is
a need to evaluate whether vehicles remain within safe bounds of state space, not just
position-space.
1.4
Thesis Overview
In Chapter 1 we focus on background aspects highlighting the need for collision avoidance
systems, especially as applied to autonomous vehicles. It is further illustrated how the
collision avoidance problem for autonomous vehicles differs from that of piloted vehicles. A
short overview of the thesis is then provided.
Chapter 2 elaborates on the basic concepts of collision avoidance as described and viewed
by contemporary researchers and focusses on the state space representation of these problems developed by Kuchar[41]. Different approaches to collision avoidance are discussed,
culminating in the probabilistic approach upon which the research detailed in this thesis is
founded. The collision avoidance problem is then compared to similar problems described
in the field of reliability theory and also to the so-called first passage time problem.
The next step is to start building on the probabilistic collision avoidance problem in
Chapter 3. It is shown how reduced uncertainty associated with autopilot intent may be
combined into a closed loop state space representation of the vehicle under autonomous
control.
The state space representation may then be used to predict state distribution
22
within a finite horizon interval through model-predictive mean and covariance propagation.
A conflict risk calculation algorithm is then designed using the predicted state distribution
and it is shown to be an intractable problem. The final section of the chapter describes how
the notion of system operating characteristics (SOCs) may be expanded upon in order to
handle the effects of time-delayed collision avoidance action.
Chapter 4 focusses on the development of a tractable algorithm for the determination
of collision risk as a function of time. The algorithm is empowered by a order reductive
collision metric and simple statistical approximations. An approximation analysis is completed and calculation errors are characterized and quantified. The chapter also addresses
a number of practical issues associated with, for example, the sizes of time horizon intervals
and simulation steps.
After development of the theory up to Chapter 4, Chapter 5 applies the research and
analyzes the performance of simulated collision probabilities as a function of time. The
research is applied to two application examples: A 2m wingspan autonomous UAV encountering a tree trunk-sized hazard; and two large commercial transports conflicting with each
other while under autopilot control.
Chapter 6 summarizes the contributions of the research and conclusions drawn, culminating in a discussion of future avenues of development and applications of the research.
1.5
Chapter Summary
It is illustrated that automatic collision avoidance systems for autonomous vehicles need to
be developed: In order to allow UAVs to operate in FAA controlled airspace; in order to
increase the effectiveness of commercial aircraft collision avoidance when under autopilot
control; and to ensure safety when piloted and autonomous vehicles are required to interact.
Differences between piloted and autonomous vehicles that influence conflict avoidance
design are discussed. The effectiveness of alerting systems for autonomous vehicles is shown
to be less sensitive to the occurrence of false alarms. Uncertainties usually associated with
23
future pilot intent can be reduced dramatically by deterministic autopilots, thereby opening
the door to more precise probabilistic conflict detection. It is the hope of this author that the
research presented in this thesis will help facilitate the implementation of more autonomous
flight systems, while making the skies safer for all.
24
Chapter 2
Alerting System Concepts
2.1
General Concepts
Kuchar [41] showed that conflict alerting systems are but a part of several safety components
in complex vehicle systems. Yang[69] illustrated how conflict detection and resolution can
further be represented in a closed loop framework. An illustration of this concept, adapted
from Yang's work, is shown in Figure 2-1. The adaptation generalizes the concept to both
crewed and un-crewed vehicles.
Together, the environment, conflict detection and conflict resolution form a collision
avoidance system. For autonomous vehicles such a system monitors the environment, predicts vehicle-relative state, issues alerts and executes an avoidance trajectory (conflict resolution). This is slightly different from traditional conflict avoidance systems where alerts
are issued, but resolution is accomplished by a human. The major elements of collision
avoidance will be discussed briefly.
First, ownshipi and air-traffic state must be monitored. Various sensors may be used
to accomplish this, but measurements are all combined in estimator architectures to provide the best state estimate and a corresponding estimate of the state distribution at time
t = to. As with most alerting systems, we always assume that the estimated state variables
are of a multi-variate normal distribution[69] [19][71]. Measurements might result in full- or
1
"ownship" is defined as the vehicle running the collision avoidance software
25
.
- -N
- .
-J1
-
--
-
'&
-
-
-
g.wAW6
010d
-
Resolution
Phase
Detection
Phase
Figure 2-1: Block Diagram of the Conflict Detection and Resolution Process
partial-state estimation.
Intent must next be established. This includes models of the auto-pilot and trajectory
scheduling systems of vehicles and also knowledge of flight plans and waypoints. We will
refer to the known flight path, before avoidance action is taken, as N, the nominal trajectory. As discussed earlier, knowledge of intent is essential when attempting to increase
prediction accuracy. A closed loop dynamic model of the vehicle must be known and when
combined with intent information we may propagate current state estimates into the future.
We next define alerting parameter(s) in order to judge whether collisions are likely.
Such parameters are known as alerting metrics or collision metrics. Part of the focus of this
thesis is in defining a collision metric in such a way as to reduce the state order, and thus
complexity, of a specific collision avoidance problem.
26
1.
After describing a collision metric, we may test it against a number of pre-defined safety
thresholds or alert criteria. For probabilistic systems we usually test for the probability of
crossing a specific metric threshold and test this probability against a pre-defined acceptable measure of collision risk. The latter is one way to ensure that the alerting system is
robust against uncertainties in the calculation of collision metrics. An example of a simple
singular alert criterium would be to alert if the minimum estimated vehicle separation will
be less than 100m some time within the next 30s (TCAS [67] [69] ).
Collision avoidance systems may decide to give an alert based on the degree of certainty
of a conflict event and also the predicted time of occurrence of the conflict. Yang assumes
that when a conflict is detected one time, it is both predicted to occur and deemed appropriate to alert an operator [69]. We will soon show how the work may be extended to
predict not only the probability of collision within a finite time horizon, but to estimate the
last possible time when it is essential to alert (and automatically avoid). The system we
propose will also provide the avoidance maneuver of choice at the time of the alert, from
amongst a pre-defined set of avoidance maneuvers.
Once an alert is issued, we may enter the conflict resolution phase. Such alerts may also
be in the form of advisories, generating an advised resolution action or trajectory. We will
especially focus on creating such a system. More precisely, we will aim to create an advised
avoidance trajectory that can be implemented by an autopilot trajectory planner. This
means that we will not only issue advisories, such as "turn left", but we will also go as far as
mapping out the specific avoidance trajectory. An autopilot may then follow this avoidance
trajectory and for crewed flight a flight director may instruct a pilot how to proceed. On a
higher level, a suggested avoidance trajectory may become the first section (maneuver) of
a newly planned flight path in a system such as that developed by Frazzoli[27 at MIT.
The above discussion provides some insight into the multitude of estimates, metrics,
thresholds and conflict resolution factors, all dependent on one another, which are found in
the literature. It is this complicated dependence which makes the problem an interesting
one, resulting in various solutions being proposed by a number of different researchers (See
survey in [43]). Most of these methods are however ad-hoc approaches, with little proba27
bilistic foundation. Varying levels of success are reported, usually under specific conditions
and for specific conflict scenarios. In this thesis we attempt to create a well-founded probabilistic approach, partially based on the work of Kuchar[42] [41] and Yang[70] [69], for a more
general avoidance scenario.
2.2
Methodologies
This section is intended to ground the research within the realm of previous research on
the topic of collision avoidance. First we show how to represent the problem within an
established framework of the field of collision avoidance and then we illustrate a method of
approach which leads up to an efficient solution space.
2.2.1
State Space Representation
In [42] and [41] Kuchar developed a state space representation of collision avoidance problems. Figure 2-2 summarizes the state space representation.
First we define the time dependent state vector
x(t) = [x1(t)... X(t)]T,
(2.1)
where n is the order of the system represented in state space and each state variable xi(t)
represents a parameter that may in some way be employed by the alerting logic. We specify
that that x(t) E R4. The state of the vehicle may change within a period of time, and the
history of x(t) within t = [ti, t 2] will be referred to as the state trajectory of the system
within [ti, t 2].
In general there are regions of, or subsets in, the state space R" that are hazardous
to the vehicle. These regions are usually defined as hazard space, sometimes denoted by
H(t). In this development we will refer to such a region as a domain of failure, denoted by
Df(t) c R", in accordance with the principles of reliability theory[5].
It is clearly undesirable to have x(t) E D 1 (t) and if x(t) enters Df(t) we say that a
missed detection has occurred.
A missed detection constitutes a failure of the collision
28
X2 *
,ex(t2 )
State
Trajectory
Figure 2-2: A 2-Dimensional State Space Representation of the Avoidance Problem
avoidance system.
The dynamics, state and relevant avoidance trajectories of a vehicle require that an
alert be issued reasonably far outside of Df (t) so as to ensure that a missed detection
does not occur.
We therefore define the alert space as Da(t), such that D1 (t) C Da-
The boundary between Da(t) and Dc(t) (the complement of Da(t)) represents the alert
threshold. When x(t) E Da(t) we issue an alert in an attempt to avoid a collision, i.e.,
avoid x(t) E Df(t). It is therefore prudent to alert well enough before entering D 1 (t) in
order to resolve a predicted conflict. When vehicles are operated by humans, we have to
account for avoidance action delays, caused by the operator's reaction time. Autonomous
vehicles will react more predictably and probably also with less delay.
29
2.2.2
Standard Definitions and Framework
At this point it is useful to specifically define a number of different outcomes to a conflict scenario. These are illustrated in Figures 2-3 and 2-4. The outcomes are split among intuitive
boundaries, and have become a standard in the field of conflict avoidance[42] [68] [30] [20].
Yang[69] and Kuchar[41] describe the definitions and framework in a similar way.
CD
FA
MD
CR
Figure 2-3: Flowchart of Different Alerting Outcomes
For conflict prediction we first need to establish if a vehicle would inevitably enter Df (t)
if no action is taken.
If no collision will occur, issuing an alert would constitute a false alarm (FA). In such a
case, taking avoidance action that induces collision also constitutes an FA. Refraining from
issuing an alert, the correct action to take, is referred to as correct rejection (CR).
When however XN(t) E D1 (t) will inevitably be true if no action is taken, then we should
issue an alert to avoid collision. Issuing an alert in time to avoid the collision constitutes
30
a correct detection (CD). Any other eventuality, such as failing to issue the necessary alert
or alerting too late, constitutes a missed detection (MD).
The only truly positive outcomes are correct detections and correct rejections. False
alarms may result in a loss of confidence in a system and, even worse, induced collision.
Missed detection always results in entrance onto Dj (t) , usually because of taking late avoidance action, since Df (t) C Da (t). State uncertainty allows us to express these outcomes
only as probabilities. For example, P(CD) would be the probability of correct detection
and similarly for P(FA), P(MD) and P(CR).
The collision avoidance problem may now be stated as follows: Given a nominal trajectory
XN(t),
a domain of failure Df(t) and J avoidance trajectories xA, (t), find an algorithm
that minimizes a balance of both P(MD) and P(FA).
In other words, the goal is to create an alert space which strikes the most effective balance
between P(FA) and P(MD). For example, if Da(t) is enlarged so that fewer missed detections occur and P(MD) is reduced, this results in an increased number of false alarms and
P(FA) will be increased. This is a conservative approach where alerts will be issued more
often. Conversely, reducing the size of Da(t) will decrease the number of false alarms and
decrease P(FA), while increasing the number of missed detections and increasing P(MD).
A fundamental and interesting trade-off thus results between P(FA) and P(MD) which
Kuchar showed is commonly encountered in signal detection theory[42][41]. In particular,
there will always be uncertainty in the outcome of alerting decisions, and a tradeoff between
P(FA) and P(MD).
2.2.3
Methods of Approach
Traditionally, collision avoidance is solved through prediction of conflict events by means of
single path and worst case approaches. Cited examples of these approaches are primarily
taken from a survey paper by Kuchar and Yang [43].
The single path approach assumes that a vehicle follows a fixed trajectory, often a
straight line, and that conflict occurs when a safety buffer around one vehicle is projected
31
X2
A
D .
Df
State
Trajectory
.
Avoidance
Trajectory
x (r)
Correct Detection
x2
X2
A
x1
A
/
x
xI
xI
False Alarm
X2 A
X2 A
x1
1
Missed Detection
X2
A
x(t
Correct Rejection
xi
Figure 2-4: Possible Different Outcomes of an Alert in a State-Space Representation
32
to intersect with D1 (t). This is a deterministic approach with a definite hit or miss result.
Some examples of this approach may be found in [40], [4] and [21].
The worst case approach propagates a vehicle's state trajectory according to its maximum dynamic capabilities. An alert is issued when it is predicted that x(t) E Dj(t) will
occur within a time frame of interest. This is still a deterministic approach with definite
hits and misses being predicted. The method was originally developed in an attempt to
model the probabilistic nature of heading estimates. It is most useful for short durations
of conflict prediction, since trajectory deviations may be severe when projected at maximal
rates. Examples of this approach may be found in [59] and [26], among others and also includes the application of viability or "reachability" theory as applied by Teo and Tomlin[62].
Most single path and worst case approaches may be implemented in real-time, since this
is partly the purpose of their creation. These are tractable, approximate, deterministic solutions to a generally intractably probabilistic problem. Such methods do however suffer from
inaccuracies brought on by approximation. These inaccuracies often manifest themselves
as sub-optimal false alarm and missed detection rates and unpredictable behavior. Most of
these systems are applicable to very specialized collision avoidance problems [43] [69] and
are highly sensitive to a vehicle's control strategy and dynamic model.
The probabilistic approach forms a middle ground between single path and worst case
methods. In this case, we determine the likelihood of collision as the likelihood of x(t) E
Df(t) within a given time horizon. Yang [69] views it as weighing all possible trajectories
with their probability of occurrence and then adding up the contribution of those that cross
into Df (t). The probability of collision accumulated over the interval [to, t], as a function of
time, will be denoted by Pc(t). In this way the amount of collision risk accumulated from
the start to the end of the time horizon, i.e. within the interval t = [to, th], becomes Pc(th).
The reason why the probabilistic approach has been used very sparingly is because, in
general, the problem is intractable[26] [70]. The single path and worst case methods were
created as approximate solutions, in an attempt to avoid having to solve for probabilities
of collision. Yang and Kuchar [69][42][70] and Paielli and Erzberger[49][50] have been at
33
the forefront of re-stating the problem in its probabilistic form and this thesis partly builds
upon, and expands the scope of, their work. Other probabilistic approaches include [66],
[54], [58] and [33].
2.3
2.3.1
A Probabilistic Approach
Concerning Alert Space
The probabilistic approach essentially redefines the notion of alert space. In this section,
we show how a probabilistic alert space results and that entry into this space is equivalent
to crossing the alerting threshold (see Figure 2-6).
In Section 2.2.1 we explained how the alert space Da(t) can be defined as a region of
state space where alerts are issued so as to avoid the domain of failure Df (t). Much of the
development effort of collision avoidance systems such as TCAS is expended in an attempt
to define Da(t)[56] [20]. In this process it soon becomes clear that Da(t) needs to be tailored
to specific scenarios and approach geometries in order to ensure that both false alarms and
missed detections are minimized. The shape of Da(t) is usually optimized across multitudes
of representative conflict scenarios, and for various vehicle applications. We will refer to
this operation as the creation of pre-compiled alert spaces.
A pre-compiled alert space ensures that relatively little processing is required to perform real-time alert calculations. Estimation of the most likely trajectory of a vehicle is
required and is then checked for intersection with Da(t). Pre-compilation needs to be a
robust solution, representing in some sense the global optimal Da(t) across a number of
different vehicle platforms and for different conflict scenarios. It is this robustness that also
curtails its performance, settling for sub-optimal instantaneous relative values of P(FA)
and P(MD). Such systems usually err on the side of false alarms, since these are usually
considered to be safer than missed detections.
Another way to view the problem is to define two trajectories: The nominal trajectory
without avoidance, denoted by N, and the avoidance trajectory, denoted by A. Then let
us also define the probabilities of collision along each of these trajectories as Pf
34
and PA
respectively. In this way we may define
P(UA)
1
- P
(2.2)
P (S A)
1 - PA
(2.3)
and
as the probabilities of unnecessary avoidance and successful avoidance respectively. Figure 2-5 illustrates the two concepts.
YES
NO
Follow
Nominal
Trajectoiy
Follow
Avoidance
Trajectoiy
Successful
Unnecessary
Avoidance
(SA)
Avoidance
(UA)
Figure 2-5: Flowchart Representation of Successful Avoidance and Unnecessary Alerts
It is useful to view a truly probabilistic approach from a different perspective. Instead
of defining Da(t) in state space, we may define it in terms of probability space, as illustrated
in Figure 2-6. The remainder of this section is devoted to illustrating this concept. Simply
put, an alert may be issued when it is deemed probable that a collision would occur along
the existing (nominal) trajectory, while there still remains a high probability of effecting
safe collision avoidance.
35
Kuchar[41] [42] shows how decision theory may be used to define an optimal region within
the space P(SA) vs. P(UA) in an attempt to homogenize the performance of an alerting
system over all encounters. P(SA) vs. P(UA) maps out a system operating characteristic
curve[41] as a function of time. This concept is discussed in more detail in Section 3.3 and
is briefly illustrated in Figure 2-6.
We may thus decide on acceptable levels of false alarms and probabilities of safe avoidance for a specific vehicle and then alert when such thresholds have been transgressed. Such
thresholds are represented by T 1 and T 2 in Figure 2-6. Probabilities of collision are then
determined using Df (t), a current state estimate and knowledge of our vehicle's dynamic
model and tested against the aforementioned thresholds of P(SA) and P(UA). P(SA) and
P(UA) therefore plot out a curve over time. Even though Da(t) does still exist, we do not
attempt to create this ever changing space. We rather make use of the fact that the region
inside Da(t) represents a certain safety risk and then ascertain whether we are inside or
outside of this risk space.
We will soon show that the main objective of this work is to find Pc(t) through application of a more rigorous probabilistic approach. Kuchar's work may then be applied in order
to determine the necessary alerting thresholds.
2.3.2
Similarity of Collision Avoidance to Alternative Problems
The probabilistic collision avoidance problem can be likened to a number of other research
interests in many different fields of study. We will briefly discuss two of these fields, since
some of the underlying concepts used to create the approach described in this thesis are
borrowed from them. The similarities of the problems are also uncanny and this realization
leads to a useful perspective.
Reliability Theory
Reliability theory is employed by a wide range of disciplines, not the least of which is the
field of civil engineering[5]. We will discuss how reliability theory may be likened to the
collision avoidance problem along the lines of a civil engineering problem.
36
-
-
-.
~
*~
-
~
1.0
-d
1
T
-.
*
-
P(SA)
Increasing time
4.4
I
(0,0)
...-
T2
Probabilistic Safe Space
P(UA)
1.0
Figure 2-6: An Illustration of the Probabilistic Alert Space
Say that a building is constructed using a multitude of steel and concrete pillars and
beams and that these serve as its major load bearing elements. Even though these elements are constructed to exacting specifications, their compression and tensile strengths
are characterized in terms of probability distributions. Given that members of different
distributions are distributed randomly throughout the construction process, we wish to determine the risk of catastrophic failure due to wind load within the next one-hundred years.
Wind speed is usually described as an "Extreme Type" I or II distribution [5].
This is by no means a simple problem and clearly very similar to that of determining
collision risk within a finite horizon time window. The problem is solved by creating a
state vector containing such information as the locations of load bearing members, the
force applied by the wind, soil density and many other relevant parameters. A safe region
37
within the state space is then defined, called the domain of safety D, and the complement
is referred to as the domain of failure Df = D'. In this case it is the domain of safety that
is bounded, as illustrated in Figure 2-7.
X2
D
S
x1
Figure 2-7: A 2-Dimensional Representation of Hazard Space in Reliability Theory
Given the state probability density function (PDF) fx(x), the probability of catastrophic failure is
Pf =1 -
LED, fx(x)dx.
(2.4)
It is from reliability theory that we borrow such concepts as the domain of failure and
some of our notation[5]. It is however no trivial exercise to find fx(x), as is the case with
collision avoidance.
The collision avoidance problem is more complex though, since an
aircraft does not only possess a state distribution, but also a non-trivial state trajectory.
Part of our method of solution, but especially the point of view adopted in this thesis, stems
from dividing the collision avoidance problem into a multiple number of reliability theory
problems, one at each simulation time step.
38
The First Passage Time Problem
The first passage time problem was first developed to describe Brownian motion[10] in
particle physics. Physicists are interested in understanding the motion of a molecule under
the influence of uncorrelated inter-particle collisions. The collisions are usually modelled as
transfer of momentum from a Wiener excitation process, and the major interest, as applied
to a single dimension of a homogeneous process, is to determine the probability of the
molecule remaining within an interval (a, b), or
a < x < b,
(2.5)
where a and b are absorbing bounds, illustrated in Figure 2-8. An absorbing bound is similar to the boundary of Df, since entry into this domain negates the possibility of exit from
the domain. The developments in Chapter 3 further clarify this statement.
Here we are dealing with a system described by an Ito stochastic differential equation
of the form
dx(t) = A[x(t),t]dt +
B[x(t),t]dW (t),
(2.6)
where W(t) is a Wiener process. In Chapter 3 we show how the collision avoidance problem
can be described by a multi-dimensional form of Equation 2.6.
Solving for the probability of leaving the interval as a function of time is then similar
to solving for the probability of an aircraft entering into Df, denoted by Pc(t). This is especially evident when one bound, say b, is considered to be very far away and the particle's
initial position is much closer to a. Pc(t) can then be solved through application of the
backward Fokker-Planck (Kolmogorov) equation[28]. This solution, even in one dimension,
is of excessive complexity. Desilles[19] provides an example of how this process may be
applied to the two-vehicle collision avoidance problem.
The most interesting aspect of the first passage time representation is that it likens the
calculation of collision risk to the flow of probability space across a boundary, into a domain
of failure, Df. The risk calculation strategy in Chapters 3 and 4 is developed from the very
39
Arg
26
-k
-
~
-
-a
=
-
I
Free Moving
Molecule
Particles
7
o 0
400
0 0 00o
x
a
Absorbing Bounds
Figure 2-8: An Illustration of the First Passage Time Problem for Two Absorbing Bounds
different standpoint of reliability theory, but also results in a solution which calculates the
amount of probability flow into Df over time.
2.4
Chapter Summary
The autonomous conflict detection and collision avoidance problem is cast into a state
space problem, similar to that developed by Kuchar[41].
As with most probabilistic ap-
proaches(e.g. [69][19][71]), we assume that the vehicle state may be described by a multivariate normal distribution. This representation culminates with the introduction of probabilities of safe avoidance, P(SA), and unnecessary avoidance, P(UA), respectively.
The calculation of collision risk within a finite time horizon is shown to be equivalent
to problems encountered in a number of separate fields of research. Two such fields are:
40
Reliability theory, especially as applied to civil engineering problems where catastrophic
failures are characterized as a function of time; and the first passage time problem described
by Kolmogorov equations, or as special cases thereof, such as the Fokker-Planck equation
often encountered in physical problems and stochastic control.
41
42
Chapter 3
Expanding the Probabilistic State
Space Approach
We have shown how the collision avoidance problem may be solved in several ways, with
the probabilistic approach being the most accurate representation. In this chapter we show
how the probabilistic approaches of Yang and Kuchar [69][41] may be expanded upon in
a number of ways. At the same time, we show why the complete probabilistic approach
is considered to be intractable for real-time applications. Three major developments are
discussed, namely:
1. Real-time future conflict probing may be model-predictive instead of approximating
the trajectory as piecewise straight-line segments.
We may also treat a vehicle's
control strategy and dynamic model as time-variant. These extensions stem from the
fact that autonomous vehicles do not suffer from the same extent of uncertainty in
future vehicle intent as encountered in piloted flight. The gain lies in propagating
conditional mean and variance, thereby reducing the state uncertainty envelope. This
aspect is described in Section 3.1.
2. The use of a repeated outcome approach such as Monte-Carlo simulation in the risk
probing process suggested by Yang[69] may be replaced by a calculation approach
such as propagation of state distribution through a combination of approximation and
convolution. Such direct approaches are usually avoided because of the associated high
calculation complexity. Paielli and Erzberger avoid this complexity by, for example,
assuming constant vehicle velocity and constant state distribution over the conflict
43
interval. Section 3.1.2 instead deals with solving the problem through a more rigorous
calculation approach.
3. After detection of a potential collision, we may delay taking avoidance action until
it clearly becomes necessary. The probabilistic effect of such time-delayed avoidance
action may be incorporated to reduce the incidence of false alarms by establishing
avoidance trajectories that circumnavigate risk peaks in probability space, as described in Section 3.3. This extension builds upon the concept of system operating
characteristics (SOCs).
The methods described in this chapter operate within a finite horizon time window from
an instantaneous realtime point of implementation, i.e. from the now, denoted by to to the
horizon denoted by th, with tw = th - to being the time window to the horizon.
3.1
3.1.1
Estimation
State-Space Formulation
Model-prediction allows us to narrow down risk probabilities based on a vehicle's dynamics and control models. Truly effective model-predictive state propagation therefore begs
knowledge of a vehicle's closed loop dynamics model and also of the intended trajectory.
For these reasons, this extension should be made when a vehicle is under autonomous control along a pre-planned flight path. The research becomes especially applicable to most
commercial and military vehicles, especially UAVs, under autopilot control.
Assume that we are dealing either with time-variant linear dynamics or with non-linear
dynamics, as long as a linearized system model is available at any time step, and linearization
errors are quantifiably small1 . The following research focusses on describing statistics for a
vector valued, linear, time-variant system of the form
XOL
y
=
A(t)XOL + B(t)UFC(t) + B.(t)w(t)
= C(t)XOL + D(t)UFC(t) + D,(t)v(t)
(3.1)
'An extended non-linear estimation approach may also be used to describe state distribution for highly
non-linear systems.
44
with w(t) and v(t) denoting white process and measurement noise vectors respectively.
UFC(t) is the feedback control input to be employed during control loop closure and XOL is
the open loop state vector. We will need to estimate the vehicle's state distribution within
a finite horizon of interest. At first glance this seems like a standard state estimation problem. There are however some important issues to consider that are discussed in the next
few paragraphs.
The vehicle's autopilot, as part of the control system, needs to be included in our
representation. Assuming that optimal state estimation is achieved, we may write out the
estimator model as
XOL
=
A(t )oL + B(t)UFC (t) + L(t)(y (3.2)
= C(t)ioL + D(t)UFC(t)
with L(t) denoting the optimal filter gain. The closed loop system takes on a different form
which we will exploit in the next subsection. This form is illustrated using the example of
a standard state feedback problem as follows:
Assume that the feedback controller may be modelled as
UFC(t) = K(t)OL.
(3.3)
The complete general model of the closed-loop system may now be constructed from
algebraic manipulation and is expressed in terms of its full-state representation
50L
XOL
A(t)
B(t)K(t)
xo
L(t)C(t) A(t) + B(t)K(t) - L(t)C(t)
2OL
Bw(t)
0
w(t)
0
L(t)Dv(t)
(3.4)
In the above representation, the closed loop system state is described by x = [XOL,
XOL]T-
A more general representation might also include deterministic inputs such as trajectory
commands. Such deterministic inputs are by definition uncorrelated with the system states.
Dynamic feedback control strategies such as lead and lag networks would also add dimension
to the .closed loop state vector.
45
v(t)
3.1.2
Propagation of State Distributions
We have thus far shown that we are dealing with a describable closed loop system when
flying under autopilot control along a planned flight path. The remaining problem is of the
form
± = A(t)x + B(t)U(t) + Bw(t)w(t)
(3.5)
where Qc(t)J(t - r) = E[w(t)wT (r)], E[w(t)] = 0, E[x(O)xT(0)] = Po, E[x(O)] = 0,
E[x(0)wT(t)] = 0 and w(t) is multi-variate white noise 2 . It is essential to note that, at
this stage, U(t) does not represent feedback control, since the control loop was closed in
Equation 3.4. Feedback control is denoted slightly differently, by UFC(t). U(t) is a deterministic reference tracking input. According to Gelb[29], with U(t) = 0 Vt, the mean of x,
denoted by mx(t), is propagated by[29]
(3.6)
hXU=O(t) = A(t)mxU=0(t)
and the covariance of x, denoted by Px(t) , is propagated by
Pxp=0(t) = A(t)Pxu=o(t) + Pxu=o(t)A(t)T + Bw(t)Qc(t)B,(t).
(3.7)
The above two propagation equations need to be expanded to account for non-zero reference
inputs, i.e., U(t)
/
0 Vt. The propagation of the state mean is clearly similar to the solution
to x(t), since
mx(t) = mxu=o(t) +
j
eA(r)(tr)B(r)U(r)dr,
(3.8)
to
therefore
rhx(t) = A(t)mx(t) + B(t)U(t).
(3.9)
As before, covariance at each time step is expressed as
PxUo(t) = E[(x(t) - mx(t))(x(t)
2
E[...] is the expected value operator
46
-
mx(t))T]
(3.10)
and since subtracting Equation 3.9 from Equation 3.5 yields
±(t) -'rhx(t) = A(t)(x(t) - mx(t)) + Bw(t)w(t)
(3.11)
(note the absence of U(t)), we may also write
Pxp+0 (t) = A(t)Pxu+o(t) + Pxuo(t)A(t)T + B,(t)Qc(t)B! (t).
(3.12)
U(t) becomes the reference path xref(t), that the vehicle is commanded to follow at any
time t. The state will be expressed as a multi-variate normal distribution with mean vector
mx(t) and covariance matrix Px(t). For this application, true state estimation is not required during the horizon of propagation, since the best estimate of x is not of importance
after t = to, only the instantaneous distribution of x.
In this way, we can now predict the state distribution of the vehicle, based on models of
both the autonomous vehicle itself and its scheduled future flight path. This is, by definition, a model-predictive solution. Mean and covariance propagation is a conditional process.
Such propagation is not simply an ah-hoc approach to solving the problem of describing
the distribution of state over time, but is a natural and proven method in the field of estimation and control. It has also been employed in research by Paielli and Erzberger[49][50]
and Prandini et al[54] for collision risk calculation between piloted aircraft. The following
deffinition now arises:
Definition: fx(t)(x) is the estimated, time-variant, probability density function of the
vehicle state x(t), with t denoting time.
It is very important to realize that the PDF, fx(t)(x), as described in this section, is
not dependent on the occurrence (or not) of collisions within the horizon of interest. This
is because the described mean and variance propagation processes inherently ignore the
occurrence of collisions. We do however employ this propagation method in the reduction
process described in Chapter 4. The next order of business is to describe how collision risk
may be determined from the knowledge of state distribution (and visa-versa) and the shape
of a so-called domain of failure Df.
47
3.2
Propagating and Accumulating Risk
3.2.1
The Single-Vehicle-Single-Obstacle
Problem
Yang employs a Monte-Carlo approach using piecewise straight-line segments for trajectory models when probing the vehicle state space for possible future conflict. This was
necessitated by the calculation complexity involved when attempting to solve for all the
complicated probability distributions that arise from the trajectory model, and the computational speed required for real-time application[69] [70].
Part of this section describes how complexity arises from the dimensionality of the problem, state-time correlation and especially the conditionality of state distribution on collision
events. We recognize that the problem is closely related to the well-known first passage time
problem of Brownian motion of particles encountered in Physics, as shown in Section 2.3.2.
We will show how to perform the required calculations, but Chapter 4 describes how to
make the calculations tractable, thereby avoiding the need for Monte-Carlo probes of the
probability space. We are constrained to dealing with a two-vehicle problem. Multiple
vehicles may for example be dealt with by resolving the problem into two-vehicle pairs, but
this concept and the complexities associated with it are considered to be outside the scope
of this research.
It is clear that a two-vehicle collision avoidance problem can always be transformed into a
single-vehicle-single-obstacle problem. In addition, this obstacle can always be transformed
to exist at the state-origin. This can be shown by defining Dfa (t) C R" and Dfb (t)
c
R" as vehicle-centered domains of failure for two vehicles respectively, and the normally
distributed corresponding vehicle states with marginal density functions given by fxa (Xa)
and fxb
(Xb).
Defining the relative state difference as x =
Xa - Xb,
the marginal density
function for the relative state becomes
fx(x) ~ N(mx
where mxa and E.
- mX, EX
+ Exb),
(3.13)
denote the vector mean and covariance matrices of the vector Xa re-
spectively. Paielli and Erzberger[49][50] and Sanders[58] also illustrate this combination of
48
mean and covariance for the two-vehicle problem. At the same time the domain of failure
becomes
Df (t) = Df.(t) U Dfb(t) c R".
(3.14)
Df (t) may be transformed or translated to exist at the state origin. Let us therefore now
only solve the problem of having an obstacle of finite size, located at the state origin, called
a domain of failure Df (t) and that an equivalent single vehicle model's marginal state PDF
fx(t) (x) can be determined within a time interval of interest, using Equations 3.9 and 3.12.
3.2.2
Accumulation of Risk
The accumulation of risk over time will be explained with the help of the discrete representations in Figures 3-1, 3-2 and 3-3. Figure 3-1 is provided as an illustration of the basic
concepts of risk accumulation, and should be kept in mind when later, more mathematical,
representations are provided. The state distribution is propagated forward at discrete time
intervals, since, as we will soon see, analytical solutions to this procedure do not exist. After
every time step, the intersection between the distribution and the hazard is integrated to
provide the added risk accumulated over that time step. The section intersecting with the
hazard is cut from the distribution, and the remaining distribution is propagated to the
next time step, and so forth.
A number of interesting aspects of the risk accumulation procedure should be noted:
1. We are dealing with flow of probability space into a hazard, over a short time interval.
It is this flow into the hazard that undergoes volumetric integration at each time step
and is them added together as time progresses, to provide the total accumulated
collision risk up to that time.
2. The hazard absorbs the probability space, since such space that has already collided
with the hazard should not be propagated any further. One way to understand the
absorbtion (absorbing bound in the First Passage Time problem), is to view the state
probability density as a cloud of vehicles and to recognize that once a vehicle within the
cloud enters the hazard, it should be removed from the process of counting further
collisions. This means that we are dealing with a conditional state distribution at
49
V Trajectory
of
state mean
Initial State PDF
Figure 3-1: A Basic 2-Dimensional Representation of the Risk Accumulation Process, Illustrated With 1-Dimensional Triangular State Distributions, for Simplicty
every time interval and the implications are that: Calculation of the conditional state
PDF at every time step becomes a complex task; the probability space diminishes as
time progresses; and collision risk increases with exposure time to the hazard
We describe collision risk within the discretized interval t = [to,
tK-1],
where
tK-1 = th
is the horizon time and t = to is considered to be the present, i.e., the point in time when
we need to determine collision risk within the next t,
= th -
to seconds. We now describe
the collision probability similarly to the first passage time probability, i.e., the probability
of entering the domain of failure by the time t
Pc(k)
= tk,
=
P[x(t) E Df(t), exactly once in [to, tk]]
=
P[xto E Df(to)] + P[xt, E Df(t1)|1xto
as
Df(to)]P[xto ( Df (to)] +
P[xt2 E Df(t2)|(Xto V Df (to)) n (xt, ( D (ti))]P[(xt V Df (to)) n (x(to) V Df (to))] +
(3.15)
where xt = x(t) to shorten notation. For the continuous time case, K -+ oo and therefore
At -- 0. The continuous case is the multidimensional statement of the first passage time
problem discussed in Chapter 2. In Equation 3.15 we assume that the vehicle has not col50
r
-
-
-
w----
- - ___w
-
-
w
__
-
--
-__-7M
lided before t = to and at every time step we need to account for the probability of collision,
given that no collision has taken place up to that time.
X1
Ds
U(t) = Xr, (t )
s
Dffx
11
fX(tI )x
(x) I B00
|B00
Figure 3-2: Progression of fx, (x) IB[o,k_1 Over One Time Step from t = to, in 2 Dimensions
Solving Equation 3.15 in its given form is a complicated task, both intuitively and in
terms of calculation complexity. We will write the equation in a different form, by recognizing that we may propagate the probability density function (PDF) describing the state
space that remained outside of DI(tk) within [to, tk], denoted by
fxk(x)IB[o,k_1]. This is
the conditional form of fxk (x), given that no collision has taken place before or on t = tk.
Definition: fX(tk)(x)B[o,k_1] is the time-variant PDF of the state x(t) at the discrete
time t = tk, conditioned on the fact that no collision has occurred while t E [to, tk-1). The
conditional statement is expressed as B[0,k_1].
At time t = to, the start of the horizon of propagation, the conditional PDF is equivalent
to the marginal PDF given by
fX(to)(X) = 9Ao (X) + gBo(x),
51
(3.16)
where gAo (x) and
gBo
(x) are the segments of the PDF inside and outside of the domain of
Df (to) respectively. At time t = to +k.At, k discrete time steps later, the equation becomes
(3.17)
*X
fx(tk) (x)|Bio,k_1] = gAk (X)+g9Bk
When we write Equation 3.15 in terms of Equations 3.16 and 3.17 we can express the
probability of collision at the discrete time t
P(k + 1)
=
jxED$(to) 9AoL(xdx
+
LxEDf(tk) gAk()[dx(1
=
Pc(k)
+
xD(tk+)
= tk
xED5(t1)
-
as
gA1 (x)dx[1
Pc(k - 1)] +
Ak+
-
Pc(O) +-
xDf(tk+1) 9Ak+1(X)dX1 -
Pc(k)]
(x)dx[1 - Pc(k)].
1
(3.18)
In this way we have shown that Pe(k) may be determined through an incremental update
at each time step from t = tk_1 to t = tk, with Pc(k < 0) = 0, provided that gA" (x) can be
found for 0 < k < (K - 1).
Finding gAk (x) entails expressing fX(tk)(x)|B[o,k_1] and salvaging the segment of
fx(tk)(x)IB[o,k_1] inside Df(tk), according to Equations 3.16 and 3.17. We therefore focus
on calculating fx(tk) (x)|B[o,k-1].
At time t = to we start with the marginal PDF
fx(to)(x) and then need to find
fx(ti)(x)|B[o,o and so forth. In general we are calculating fx(t,+1)(x)|B[o,kI when
fx(tk)(x)|B[o,k_1] is known and fx(to)(x)|B[o,_1] = fx(to)(x). To do this we need to understand how Xk+1 can be realized from Xk. Assuming Euler integration, we can write
Xk+1
=
Xk + Attk
=
[I + AtA(k)]Xk + [LAtB(k)u(k) + AtB(k)w(k)]
=
Mk(xk) +Rk[u(k),w(k)]
(3.19)
where Mk(Xk) and Rk[u(k), w(k)] are independent quantities. This independence allows us
52
to apply n-dimensional convolution to find
(3.20)
fxk+1(x) = fMik (m) * fR, (r).
In the same way, we may propagate fx(t,)(x)|B[o,k_1 by setting
fX(tk+1)(x)BO
* fR
9B(X)
=
fxxtD((tk) 9Bo W
f
1f
1
XD(tk)
ftk
()
9Bk
(r)
gx
(X)gA * fRtk (r)(.1
where
By
_
1-
9B W(.2
Lk) f(tk) A(x
(3.9A22)
and with Pc(-1) = 0. In Equation 3.21 gBk(x) represents the segment of fX(t)(x)|B[_,k1]
outside of DI(tk) and fxgDf(tk) 9Bk(x) normalizes
gBk(x)
to become a true PDF, in accor-
dance with Bayes' rule. It is essential to realize that only that part of fX(tk)(x) |B[o,k_1
outside of D1 (tk) must be propagated, since fx(tk+1) (x) IB[o,kI is by definition conditioned
on B[o,k).
No closed-form representation of the convolution in Equation 3.21 is however
known and the computational complexity of a numerical solution is O(N 2n), with N being
the number of quantizations of each dimension of the space and n = dim(x).
3.2.3
Propagation Procedure
So far we have made assumptions about the nature of the system under consideration, such
as it being linear, time-varying. We are also dealing with a one-vehicle-one-obstacle problem
with normally distributed states. We have shown that this is equivalent to a two-vehicle
problem, and have described the relative state and domain of failure in a way showing similarity to that found in the work of Paielli and Erzberger[49] and Sanders[58]. Given these
assumptions and accepting that the vehicle has either undergone no collision before t = to
or that we can cast probability of collision as the probability of collision within [to, th], then
the derivations so far from Equations 3.13 to 3.21 are accurate discrete time representations.
Figure 3-3 illustrates how the exact discrete time probability of collision accumulated
53
fx(,)(x)I B[0 k 11
S Bk
k;
x
1
Df(tk
g4 (x)
gB,(W
"k;
Bk
x
Df(tk
D, (tk| D (tk )
D (tk
4
X" X
P (k) = P(k -1)+
fg.,
(x)dx[ - P,(k -1)|]
f& x
Bk-
1-
'B
Jg(x)
xeD,(rk)
Bk
Ak!
x
)
D, (t)
,x(t,l(x)|i
= fx([r $
t5
D
5
R( )r
fx(t..)(x)|IB[Ok
4
k.1
-B
k.1
x
Df(tk+)
D (tk+l
Figure 3-3: 5-Step fx(tk+1)(x)IB[ok] Propagation and Calculation of Pe(k) (1-Dimensional)
54
within [to, tk], denoted by Pe(k), may be found through a 5-step propagation process. The
illustration is provided for a one-dimensional problem in order to reinforce understanding
of the process. Notice that the propagation process requires knowledge of Pe(k - 1) in
order to find Pc(k). This implies that the 5-step process needs to be applied incrementally for each time step in [to, tic], starting at to and in increasing order of k. Recall that
fX(to)(x)|B[o,-1] = fx(to)(x) and Pc(-1) = 0. At every time step t
= tk,
Step 1: Segment gA,,(x) from fX(t)(x)B[o,k_1] by setting
fX(t)(x)IB[,k-1] = 0 V x $ D5(tk).
Step 2: Use Pc(k) = Pc(k - 1) +
fxEDf(tk)
9A,(x)dx[1 - Pc(k - 1)] to accumulate
the probability of collision.
Step 3: Segment gB from fx(tk)(x)IB[o,k11 by setting fX(tk)(x)|B[o,k_1] = 0
V x E DI(tk).
to create the PDF
Step 4: Normalize
gB,
Step 5: Propagate
ftk) (x)
convolution
fx(tk+1)(x)IB[okI
ft
(W
x
_
oED
1 (t) gAk(X)
to the next time step through the n-dimensional
=
fxk)(x) * f
Now continue the process by utilizing
fX(tk+1)
(r).
(x)IB[o,k] at time t = tk+1 to obtain
9A, (x) as in Step 1, and so forth.
The numerical complexity of the propagation phase is dominated by the O(N
2
,)
con-
volution process indicated in Step 5 and described by Equation 3.21, and is therefore an
intractable problem for meaningful values of n and N. Chapter 4 deals with order reduction
metrics and approximation techniques that make this calculation tractable.
3.3
3.3.1
Time-delayed Avoidance
System Operating Characteristic
The performance of a collision avoidance system may be characterized as a trade-off between
unnecessary alerts and successful alerts. This method of performance evaluation is based on
the use of the System Operating Characteristic (SOC) curve, developed by Kuchar[41][42].
The curve facilitates a visual exchange between P(SA) and P(UA) as the time of implementation of a single avoidance maneuver is varied. It is assumed that no collision has
55
taken place before the time corresponding to any point on the SOC. The SOC curve is always specific to the avoidance maneuver and the geometry of the encounter, see Figure 3-4.
The top left corner of the SOC figure represents perfect, safe avoidance strategies with zero
probability of false alarm. Curves that approach this corner therefore allow for improved
alerting thresholds. The opposite is true for the bottom right corner of the SOC figure.
1.0
P(SA)
(0,0)
P(UA)
1.0
Figure 3-4: Different Shapes of SOC Curves
SOCs are not created for t > to when attempting real-time conflict alerting, because of
the uncertainty in the future intent of most vehicles and the method of creating estimates
of P(SA) and P(UA). This means that an SOC curve is mapped out as time passes, but
the most futuristic point on the curve is at t = to, the present.
This point, by defini-
tion, evaluates the performance of immediately employing a collision avoidance maneuver.
Real-time collision avoidance strategies such as [70] and [69] make use of this most recent
point on the SOC curve to decide on an alert eventuality. Earlier in this thesis, we have
56
shown that vehicles under deterministic and stable autopilot control allow us to propagate
their state PDF into the future, without the severe increase in state uncertainty usually
associated with unknown pilot intent. Section 4.3.1 describes how the finite horizon time
window cannot however exceed td, a bound based on our knowledge of intent, since intent
is never known exactly within an un-bounded period of time. Therefore, for this discussion, we will predict the shape of the SOC curve for to
3.3.2
t < to+ t,
where t, = th -to
! td.
Condensed SOC
Unfortunately the shape of SOC curves do not always follow a simple path to the origin, as
usually depicted for vehicles on an inevitable collision course, seen in Figure 3-4. Various
examples of collision avoidance exist that result in more complex SOC curve representations. One such example is illustrated in Figure 3-5. In this case, a vehicle is flying directly
towards a domain of failure, and employs an avoidance maneuver that only temporarily
ventures off the nominal trajectory. The avoidance maneuvers are investigated tAD seconds
apart.
A more complex SOC curve results from the example in Figure 3-5, illustrated in Figure 3-6. Typical alerting thresholds such as (P(SA) < Ti) n (P(FA) < T 2 ) would alert at
time t = to when the curve is only known up to t < to. Without certainty of the curve at
t > to, an alert would be issued. This is the standard approach to application of the SOC
curve [41],[70]. A more complicated scenario, such as the given example, can however be
envisaged where the curve later exits the alert-space demarcated in Figure 3-6. A finite
horizon simulation is able to draw the curve up to t = th, therefore predicting whether such
an exit will occur within the finite time window.
A number of interesting questions now arise. First, should the alert be issued upon
entering the alert-space even if an exit is foretold?
Second, how long and how far in-
side the alert-space may a vehicle remain with a degree of safety? It is ventured that an
SOC curve alone cannot answer these questions, since future probabilities of collision are
time-correlated, as described in the previous section and should therefore be conditional on
previous collision risk.
57
11
1
-
---
- --
=-
u
---
%c
--
-~
-
-.-
-
a
Avtod
....
-
Avoid at
'
4 +
t=t
A
Avoid at
t=t +t
..
......................................................................
41
AD
0
-4
Avoid at
t=t
0
+3t
AD
Figure 3-5: An Example Showing the Benefit of Delayed Avoidance Action
Let us assume that a vehicle may either continue along a nominal trajectory N or employ
a single avoidance strategy A at any time t. Continuous time delayed avoidance may then be
viewed as the creation of an infinite set of compound trajectories where a vehicle continues
along, for example, trajectory N for 0 < tN < td seconds and then branches onto A for the
next 0 < tA
td - tN seconds. We now discretize the number of opportunities to transition
from N to A, and prevent the vehicle from transitioning back to N. Each transition time
is designated as illustrated in Figure 3-7. In this way, any one of the discrete compound
trajectories may be defined as
DCT = [N(, ti), A(ti, td)].
58
(3.23)
1.0
to+
2 tAD
to
P(SA)
<
I
Cd
I
o | Probabilistic Safe Space
T
(0,0)
1.0
P(UA)
Figure 3-6: The Unpredictable Nature of the SOC Curve for t > to
The avoidance discretization step size will be denoted by
steps are denoted by I
= td/tAD
tAD =
t 1 - to and the number of
+ 1. In this way we may describe (I - 1) SOCs, where
P(UA) is derived from the probability of collision of the nominal path N = DCTI, and each
other DCT within i = [0, I - 2]. The issue of the size of
tAD
is considered in Section 4.3.2.
Now we may create a new kind of SOC, the Condensed SOC (CSOC), which corresponds to that single one of the (I - 1) SOC curves with the highest P(SA) at t = to.
Notice that the P(UA) of each SOC is exactly equal at t = to. In this way, the CSOC
provides the avoidance option with the highest P(SA) while condensing the outcome of the
next
td
seconds into a single point at t = to. We no longer have to deal with the issues of
an SOC possibly entering and exiting the alert-space within a window of time, but simply
need to ask if the CSOC is inside or outside of the alert-space when deciding to alert or
not. Recall that the "alert" referred to here really is the decision to let the autopilot of
59
--
~-.
-~-
-
-Iu-;m~u
-~
- ---
~
-
-
-
Xi
Ao
t,
ti
to
X
2
Figure 3-7: Creation of Compound Avoidance Maneuvers
our vehicle implement an avoidance maneuver. Notice also that the avoidance maneuver is
delayed by ti seconds, where i* corresponds to the designation of the selected SOC that
became the CSOC. The avoidance maneuver associated with the CSOC now becomes the
suggested avoidance maneuver if an alarm is issued.
It is worth reminding the reader that the collision avoidance problem solved in this
thesis is a conservative one where only very small amounts of collision probability are tolerated. In this sense the alert-space bounded on the CSOC curve would be described by
(P(SA) < T 1 )n (P(FA) < T 2), where TI, T 2 > (1- Pc.) and where Pm,.
is the maximum
allowable collision risk that a vehicle may be exposed to within any finite horizon interval.
The alerting aspect of the decision making process does not need to know exactly where
a point on the SOC curve is, only which side of the alert-space boundary it is on. We
therefore only calculate Pc(k) until Pc(k)
Pcmax becomes true.
60
The CSOC reflects a change in philosophy from the original SOC in terms of what the
metrics for decision making should be. An SOC essentially employs the metric of "needing
to alert" vs. the "success of alerting now". In the case of the CSOC we employ the metrics
"need to alert" vs. "safety", where "safety" accounts for the possible benefit of delaying
avoidance action now, while acting later. The CSOC concept can therefore be viewed as an
extension of Yang's work on selecting the most beneficial avoidance maneuver from within
a set of avoidance maneuvers, at t = to [69].
The concept of a CSOC is created to link this dissertation with the works of Kuchar and
Yang and to be able to apply existing methods of analysis and alerting threshold design to
the time-delayed avoidance problem. The rest of the thesis, including application examples,
will however focus mainly on the calculation of Pc(t) (in the form of either P (t) or PcA(t))
so that SOCs may be created according to Equations 2.2 and 2.3. For trade-off between
P(SA) and P(UA) in alerting threshold design we refer the reader to [42],[41] and [69].
It is also worth pointing out that we do not really need to create I number of SOC
solutions before being able to find the CSOC. The principles of dynamic programming can
be employed to find the SOC with the highest P(SA).
3.4
Chapter Summary
The development in this chapter rests on three major assumptions, namely that: A vehicle
may be modelled using time-variant, linear dynamics; collision risk only extends from the
present into the future, that is, collision risk only makes sense when calculated after t = to,
the start of the model-predictive, finite horizon simulation window; and Euler integration
is used in order to time-discretize the intractable, continuous-time collision risk calculation
problem.
The autonomous vehicle problem is extended to a closed loop representation where it
is shown that mean and covariance propagation may be used to estimate future vehicle
state uncertainty through finite horizon model predictive simulation. This development is
a direct consequence of the reduction of uncertainty of pilot intent because of autonomous
61
control. Paielli and Erzberger[50] and Prandini et al[54], among others, also illustrate a
similar development, but for straight line vehicle trajectories.
The calculation of collision risk within a finite and discretized time horizon is cast as
a reliability theory problem and developed from first principles. The solution is developed
as a set of discrete propagation equations illustrated in Figure 3-3 and it is shown that the
calculations are numerically intractable when real-time solutions are required.
The notion of the system operating characteristic (SOC) [42] [41] employed as a trade-off
between P(SA) and P(UA) when making alerting decisions is extended. The SOC at any
point in time is based on the outcome of making an avoidance decision at that time. It is
shown that the effects of time-delayed avoidance action may be incorporated into the SOC
representation, resulting in a best-case or time condensed SOC (CSOC), for a specific time
horizon and a given set avoidance trajectories. This allows us to make use of signal detection
theory in order to optimize alerting thresholds, while at the same time investigating the
pay-off of delayed avoidance action. As a direct consequence, the work of Kuchar may be
employed when optimizing alerting thresholds[41].
62
Chapter 4
Tractable Risk Propagation
Through Quadratic Collision
Metrics
4.1
Order Reduction of Convolution
The intractability of the n-dimensional numerical convolution required to propagate a vehicle's state PDF arises mostly from the O(N 2") computational complexity involved in
solving Equation 3.21, where N represents the number of samples along one dimension of
the discrete numerical solution grid. At first it also seems that the n-dimensional integration of Equation 3.18 could potentially add additional O(N") complexity to the solution.
Fortunately this integral may be solved through a gradient search approach of O(nN)[5].
In the event of no collision risk within [to, th], no convolution would be required and the
mean and covariance propagators described in Equations 3.9 and 3.12 would provide an
optimal estimate of the vehicle's state PDF at any time within the horizon. Unfortunately,
resolving fx(t,)(x)|B[o,k_1] into gAk(x) and
Bk
(),
respectively, adds complexity. This is because only
segments within and without of Df (tk)
ftk) (x) derived
from g,
(x) (see Equa-
tion 3.21) must be propagated forward in time and it is no longer a normal distribution
for which special properties might be exploited to negate the need for the application of
n-dimensional convolution. Hence, analytical solutions to Equation 3.18 and 3.20 are un-
63
known. This chapter focusses on the application of intuitively natural collision metrics in
order to reduce the O(N
2
,)
complexity of risk calculation to O(Nlog 2 (N)) through system
order reduction to n = 1 and the application of efficient numerical convolution techniques.
4.1.1
Simplifying Assumptions
Existing collision avoidance systems such as TCAS define a so-called Distance Modification
(DMOD), a collision safety bound (distance) [67] [48] around an aircraft which corresponds
to a radius of conflict. TCAS then predicts the amount of time it would take any vehicle to
close to within DMOD. It then issues a Traffic Advisory (TA) when it is predicted that an
aircraft will close to within DMOD, within r1 seconds. A second level of alert, a Resolution
Advisory (RA) is issued when 72 <Ti1 becomes the predicted time to conflict. The DMOD
distance essentially defines a equivalent domain of failure around a vehicle.
It is in the spirit of TCAS, its well-established track record and its wide acceptance as
a standard for comparison[67] [41] [69] [48] [24] [25] [56] that we choose a similar safety bound
or DMOD surrounding a vehicle. Many other systems with this approach are cited in [43].
It will be shown that this tried and tested bound may in fact be shaped into a useful order
reductive metric. We also generalize the metric to shapes with varying radius around the
vehicle, by defining it as a quadratic function of the vehicle's state described by
R=
x-TBx,
(4.1)
where B is constrained to be a real, symmetric positive definite matrix.
The square of the R metric is more useful, since the square root operator in Equation 4.1
spawns considerable numerical and analytical complexity when dealing with the calculation
of PDFs. The metric therefore becomes:
- = R2
=
XT Bx.
(4.2)
For typical spherical buffers around vehicles where position is a simple state, B only retains
some unitary diagonal entries. The focus of the next section lies in reducing the required
64
convolution order of the n-dimensional state distribution by applying this one-dimensional
metric. Soon we will need to find the probability distribution of y and the lack of a square
root operator in Equation 4.2 makes this feasible. The collision safety bound (the boundary
of Df(t)) of an aircraft must be expressed in the form of Equation 4.2 in order to apply the
algorithms developed in this dissertation.
4.1.2
Mathematical Reduction
A Simplified Propagation Equation
Define -yo as the safety buffer threshold (the square of the DMOD) or the boundary of Df,
i.e., if
(4.3)
^
7- 7o,
then collision has occurred. Equation 4.3 now becomes the reductive collision metric of
interest, since we are dealing only with a one-vehicle-one-obstacle problem as described
in Section 3.2. We aim to determine the probability of satisfying this inequality within
[to,
tk],
which is equivalent to evaluating the probability of collision, Pe(k). The full-state
representation of Equation 3.19 may be replaced by the single-state equivalent
Yk+1
=
k +Atik
=
Yk + Ak,
(4.4)
where Atyk will be referred to as Ayk, simply to avoid unnecessarily complicated future
notation issues. In addition, it will be shown at a later stage that Aik can also be conveniently expressed as a quadratic form of x (see Subsection 4.1.3).
The simple summation of random variables in Equation 4.4 leads us to also express -yk+
as a random variable. In order to do this we recognize from Figure 4-1 that the cumulative
distribution function (CDF) of 7k+1 is described by
Fr
(+ )
Pk+1 <
B~osk).
k]
In this way, the marginal PDF of yk+1 may be described exactly by[52]
frk+(y)|B[,k
d
+o
= d-
d-i 1i
-o
65
B
f1i
q,$(,#)dqdV),
(4.5)
Atk
Ak
=k+
Yk+1
~
Yk +Atk
J.k+
t
. . . . . . .
....
F:F
0
Figure 4-1: Region in the 7yk-Ati Plane where
T,
yk+1
rk
Yk+At$k,
Used to Describe Fr+l
C-)
where
fBk
(,A
rk)ALk
FBk
d-ydA
rPk,Ark
(
(4.6)
is the joint PDF of -/ and A4 at t = tk, given that no collision has occurred within [tO, tk].
Also, 0 and
4.1.3
#
are simply variables of integration.
Simplifying Approximations
The reductive metric results in Equations 4.4 and 4.5, which are simpler to deal with than
their full-state n-dimensional counterparts. There are however two issues that cannot be
circumvented without approximation:
1. The double-integral of Equation 4.5 needs to be approximated numerically with complexity of O(N 4 ), unless y and A4 are independent, in which case complexity of
O(N 2) results.
66
2. No solution is known to the joint PDF in Equation 4.6. Numerical approximation
would be of O(N 2n+ 2) complexity, since we would need to revert back to the fullstate representation and employ e.g. a Monte-Carlo method to solve.
Approximation will need to fulfill at least two primary requirements:
1. The calculation order complexity of Equations 4.5 and 4.6 needs to be reduced significantly. Most digital signal processors (DSPs) are optimized for O(N 2) or less
complexity, since this is widely considered to be the feasible extent of tractability
and applicability for image processing applications.
We will show how such DSP
techniques may be used to solve our risk calculation problem and therefore aim at
reducing our complexity to O(N 2 ) or less.
2. Approximation errors need to be characterized to a sensible extent. This would imply
that we need to estimate or exactly determine the sizes of all sign-indefinitel errors.
Sign-definite 2 errors at least need to be clamped between upper and lower bounds, with
an emphasis on the calculation of the upper bound. Upper bounds on, for example,
Pc estimates translate into reduced estimates of P(SA) and P(UA) and therefore lead
to a more conservative and usually safer alerting approach for autonomous vehicles.
It is clear that the major issue of complexity is the calculation of Equation 4.6. Recall
that Equation 3.15 in the previous chapter provides the discrete representation of Pc(k) =
Pc(tk). The aim is to find an upper bound to the calculation performed there-in that would
provide an equivalent expression to Equation 4.6, but of considerably reduced complexity.
To this effect we approximate
Pc(k)
=
P[x(t) E Df (t), exactly once in [to, tk]]
=
P[x(to) E Df (to)] + P[xt1 E Di(ti)Ixto ( Df(to)]P[x(to) ( Df(to)] +
P[Xt2
e
P[x(to) E D1 (to)] + P[xt1 E Df(ti)Ixto V Df(to)]P[x(to) ( Df(to)] +
P[xt 2
=
E Di(t 2 )|(xto ( Df(to)) n (xti ( D1 (ti))]P[(x(to) ( Df(to)) n (x(to) V Df(to))] +
E Df (t2)|xti V D1 (ti)]P[x(ti) V Dje(ti)] +
P[x(t) E D1 (t), at least once in [to, tk]]
'Neither positive or negative definite, i.e., the sign is unknown
2
Either positive or negative definite, i.e., the sign is known
67
(4.7)
in accordance with such texts as [65],[61],[60],[15],[16] and [64].
This provides an upper
bound approximation to Pc(k), since
P[x(t) E Df(t), at least once in
[tO,tk]]
> P[x(t) E D 1 (t), exactly once in [to, tk].
(4.8)
The approximation of Equation 4.7 allows us to employ unconditional means to calculate
Equation 4.6, such that
(,
=
rkAt
U y - 7o)frAI (7,Ai)
-to frAN (7 , A)dydAd
(4.9)
f_+
where u(y - 70) is a unit step function and frkAfk (y, A)
is the joint PDF of 7 and Ay at
t = tk, regardless of any collision activity having taken place within [to, tk].
1-
f+00 fo'0 frkAk (Y, A)d-ydAS
is simply a scalar normalization according to Bayes' rule.
One more approximation is made, this time to reduce the complexity of Equation 4.5:
Assume that -yand At are independent within any one time step
note that this is very different from assuming than -y and
[tk, tk+1].
It is essential to
are independent within [tO,
th],
as will be shown shortly.
Our independence assumption and assumption on the calculation of Pc(k), combined
with the application of Leibniz's rule [52] allows us to re-write Equation 4.5 as
fr+1 (7r)|B[kk]
frk+1(7)IB[o,k]
=
fj_()
*
fAN(M,'
(4.10)
where
p Bk(_
u( Y 7o)frk(7)
1-f0frk(7)d-y
and frk(7) and fAt (Ai) are the marginal PDFs of
7 and Ay respectively, without con-
ditioning on any collision event. The usual normalization factor is again present in the
denominator.
68
It is Equation 4.10, a one-dimensional convolution, combined with the ability to operate
on the unconditional marginal PDFs of
y and At to find Pe(k), that makes the solution to
this problem tractable. It is shown in Section 4.1.3 that the required marginal PDFs may
be calculated for a quadratic form of normally distributed state variables.
Figure 4-2 shows how to go about solving for Pc(k) at each time step t = tk. This figure
may be contrasted to the exact full-state calculation illustrated in Figure 3-3.
Assuming that the unconditional marginal PDFs of -y and A- are known, the process
of finding Pc(k) and propagating fr(t,1) (-)|B[k,k] may be divided into the following steps:
Step 1: Find gAk
(7),
the segment of frk(y)|B[k_1,k_1 for which y :5 yo.
Step 2: Make use of Pc(k) = Pc(k - 1) +
fO"
gAk(y)dx[1 - Pc(k - 1)] to accumulate
the probability of collision.
Step 3: Apply Equation 4.11 to find
frk
(-y). Note that this step does not rely
on steps 1 and 2.
Step 4: Apply Equation 4.10 to propagate
Bk
fir,
(-Y)
into fr. l1
(7)|B[kk].
) Bk J
Now continue by returning to Step 1 for calculation of Pc(k + 1) and so forth.
A clear segmentation between the above outlined steps is evident. This is because Equation 4.7 allows us to express Equations 4.8 and 4.11 as functions of unconditional marginal
PDFs that are independent of the conditional operations at the previous time t = tk_1. The
conditional PDF, frk+1(y) IB[k,k], in Step 4 is created by assuming independence between
y
and Ay within only one time step. This segmentation leads to a very useful result: Even
though errors resulting from the independence assumption do accumulate in the calculation
of Pc(k), these errors are not propagated into the creation of frk+1 (Y)|B[k,k] at any next
time step t = tk, thereby resulting in a convergent rather than divergent approximation of
Pc(k) as t -+ th.
An alternate view of the apparent segmentation is that the independence assumption is
only employed over the course of one time step and only to determine the amount of flow of
probability into D1 (tk). The seemingly simpler route of assuming independence within the
entire interval [to, th] and using it to propagate frk+1 ('i)IB[o,k) through the resulting simple
69
frI)(Y)
I
B
k
1
Df(tk
Bk
D (tk
.0
gA(y)
-~
2
P,(k)= P,(k -1)+
U(7~r(t)()'
3
fg, (,vdy[1 -P, (k- 1)*
0
2'
Df(tk ) Ds (tk)
fr(k)|
B[k k =
Op
frA)
fr(")(7)|Bf't(k
4f
Assume fr(tk (Y) and frrtk)(A)
known over
IV'
[to , th]
Df(tk+ )
(tk,1
Figure 4-2: 4-Step fr(tk+l)(y)IB[k,k] Propagation and Calculation of Pc(k)
70
one-dimensional convolution, is folly, since convolution always leads to an increase in the
variance of the PDF. In reality, the variance of a quadratic metric should reduce drastically
when its mean approaches yo, because the chi-square distribution narrows when it nears
the origin.
Errors resulting from the two approximations made in this section are investigated in
Section 4.2.
Methods of Solution
The focus now becomes finding a solution to frk (7) and
fAt, (A) within [to, th]. We will
only show how to solve for frk (y) and eventually prove that
fA,
(Ay) may be solved in
the same way.
Other than the useful special cases discussed in Subsection 4.3.3, the PDF of a general
quadratic form of normally distributed random variables needs to be approximated. Methods of approximation include series expansion, numerical Fourier inversion, Monte-Carlo
and particle-based filtering simulations and approximations based on moment matching
techniques. Moment matching through processes such as Cornish-Fisher,Gram-Charlier
and Edgeworth Expansion[35] [45] [6] do not provide accurate results when tails of PDFs are
of value and provide little or no guarantee on rates of convergence relative to the number of
cumulants matched[35]. In this case Monte-Carlo techniques are computationally expensive
when exploring high-dimensional probability spaces, again because of the high accuracy
required when representing the tails of estimated PDFs, as with most problems in risk and
reliability theory. It is partly for this reason that Yang [69] also performs a dimensional
reduction to vehicle heading, before attempting Monte-Carlo simulation.
Fourier Inversion emerges as the salient choice, since it turns out that the inversion
process can be greatly simplified for the quadratic collision metric. When applying Fourier
Inversion we would first need to solve for the marginal characteristic function of y given by
Ork
(r)
+0 .
r0o
.
= E[ei"] = 0 e"1fr (7)d-y
71
(4.12)
and then solve for the FourierInversion Integral (FII)
fr (Y) = 2r
(4.13)
00 #rr(r)eiw'ydr.
_oo
Characteristic Function
Solving for a characteristic function can in itself be a daunting task. The next few para-
#(r) does
graphs however show that a closed form solution to
exist in an analytical form,
when dealing with the proposed quadratic metric.
First we transform y into an appropriate form using the substitution y
=
(x - mx)
where mx is the vector mean representation of x. In this way we find that
y
=
xTBx
=
m TBmx
=
E + ATy +
+ 2mTxBy + YTBy
(4.14)
1TMy.
When we solve the generalized eigenvalue problem such that
... ,
where A = diag(A,
CCT
=
CTMC
=
X
(4.15)
A
An), then we can substitute y = Cz and
6 = CTA into Equation 4.14
such that
-y
=
=
e+sTz+!zTAz
n
1
) + E(iz; + 2Azf)
(4.16)
i=1
where z = (Z1,
reshape
... ,
zn)T is a standard normal vector and 6
=
#r,(r)into
#r,(T)
=
E[e T -]
1 +00e'
-oo
72
d
fr u(h
(61, ... , on)T E Rn. Now we
=
and recognize that E[ei'r(7-)] =
e-reE[ejr(-E)]
X2 (r) where
#2(r)
(4.17)
is the characteristic function of a
2
non-central Chi-square (x ) variate in z, where z is in standard normal form.
known analytically[34] and when substituted into Equation 4.17
#r (r) = e.rE
JJ
1
e_
#rk (-)
2
5?2 /(1-3Ajr)
i=1 v1 - j3Te r.8
#X2
(r) is
becomes
(4.18)
an analytical form to be substituted into Equation 4.13, requiring no numerical approximation. The computational complexity of function evaluations in Equation 4.18 is 0(nN)
and that of the eigenvalue decomposition (Equation 4.15) is in general O(n 3 ). The latter
may be reduced to O(n), as described in [18].
Fourier Inversion Integral
Various options exist when attempting to solve the inverse Fourier integral (Equation 4.13)
using the characteristic function in Equation 4.18. The first is to perform brute force numerical integration and to reduce the discrete step size of t and w until the error on the
calculated distribution of -y is negligible. This is not difficult but requires computational
cost O(N 2 ). Care must however be taken to reduce the error of this calculation to be negligible when compared with the small amount of allowable maximum probability of collision
defined in the previous chapter. In this case, the key to error analysis of Equation 4.13
because of discrete integration is the Poisson summation formula[34].
The second numerical method of calculation is more powerful and less intensive and
amounts to the use of FFT algorithms in order to reduce computational cost from O(N 2)
to O(Nlog 2 (N)). As is the usual practice in engineering, we define the Fourier Transform
(FT) as [55]
e- 22uf(x)dx.
F: f(x) F-+ F(u) =
Substituting u
=
(4.19)
in Equation 4.19 we can show that
F(u) = #rk(-21ru).
73
(4.20)
We now attempt to find f(x) = fr, (7) by Inverse FourierTransform (IFT), which is clearly
equivalent to applying Equation 4.13, the FII. All numerical solutions to IFT are approximate, unless f(x) is continuous in x, periodic with period T and band limited to frequencies
Jul < N/2T where the integer N represents the number of discrete samples. We do not
expect f(x) to be periodic, since, by definition,
f_.
f(x)dx = 1.
Hughett [32] shows that FFT approximation of f(x) can now be accomplished by first
splitting it into f(x) = fc(x) + C, where fc(x) is absolutely integrable and C is a constant.
This split is done to avoid singularity problems when applying the FT. We now focus our
attention on
00
fc(x) =
(U)ej2nxdu
(4.21)
J-00
which can be approximated as a sum of rectangles with width 1/T to obtain
00
fc(x) = 1
F9[k]ej2 ,rkx/T,
(4.22)
k=-oo
the FourierSeries (FS) representation of fc(x), with
(4.23)
Fs[k] = Fe(k/T)
T
Both Proakis [55] and Hughett [32] show how the solution of FS through
N/2
f(x)
g(x)
G[k]ei 2 ,rkx/T,
(
=
(4.24)
k=-N/2
where G[k] is defined by
for k = 0,
F(O)/T +C
G[k]
=
for 0 <
F(k/T)/T
0
Ikl
< N/2,
(4.25)
for |k| = N/2,
provides the most accurate known numerical approximation of f(x). Notice that Equation 4.24 is usually defined [55] as the Inverse Discrete Fourier Transform (IDFT) and that
it may be solved using the Radix-2 Inverse Fast Fourier Transform (IFFT) algorithm, as
long as N can be written in the form N =
2E
74
where E is a positive integer. IFFTs are
accomplished with at most N/21og 2 (N) multiplications and Nlog 2(N) additions, thereby
reducing computational complexity to O(Nlog 2 (N)).
Concerning A
In previous sections we have shown that frk (7) can be calculated within [to, th], since - can
be converted into the required quadratic form. It turns out that
ALk can also be expressed
in this quadratic form as shown in the following derivation, where B is symmetric:
A
=
XTBx
=
AtjTBx + AtXTBc
=
2At&TBx
=
2At[Ax + Buu(k) + Bew(k)]TBx
=
2AtxTATBx + 2AtuT(k)BujBx + 2AtwT(k)BT Bx
X TNx+Sx,
(4.26)
with N = 2AtATB and S = 2At(uT(k)BTB + wT(k)BTB), and where At is the time step,
such that
At = tk+1 - tk-
Equation 4.26 would be in a quadratic form similar to Equations 4.2 and 4.16, but for
the term 2AtwT(k)BlBx which contains the random variable w(k), white noise. This single
term would make the dimensional and computational reduction discussed so far intractable
again, unless we impose the constraint that
B B =[0].
(4.27)
At first glance this constraint seems excessive, but it simply means that no states that form
part of the calculation of -y may be directly excited by white noise. In other words, B
may not contain non-zero terms which operate directly on states excited by white noise.
Collision bounds on position (as is usually the case) would require that position and velocity be part of the state vector. Even the simplest Newtonian models would include white
noise excitation at acceleration or jerk level, not velocity and position. This means that
the constraint usually holds for typical vehicle models. If, for some reason, the constraint
75
does not hold naturally, white noise can be shaped with a high bandwidth low pass filter in
order to add another state buffer between any state required in the calculation of Ay and
the noise excitation.
within [to, th] from the PDF of x described by
We may therefore calculate fAr k (A)
propagation of Equations 3.9 and 3.12. This is done in the same way as outlined for fr(y)
in the preceding sections. Also, recall that Equation 4.2 was created for B belonging to the
class of real, symmetric, positive definite matrices. A
is clearly not sign-definite, since -y
may increase and decrease in value. The only constraint on B employed in the calculation
of fr(y) is that it must be a real matrix. Symmetry and positive definiteness were only
exploited while finding Pe(k) and proving that At is a quadratic form.
Lastly, in Section 4.2 we require that A be written exactly in the complete square form
described by Equation 4.2 when calculating the covariance between y and Ay in Equation 4.39. The covariance calculation affects the estimate of calculation error because of
the independence assumption. It is not immediately obvious that we may always write Ad
in the required form and this will not be proven true or false. It will however be stated
that Ad
can be written in the complete square quadratic form whenever the only non-zero
elements in B are on its diagonal and all the derivatives of states operated upon by non-zero
entries in B are themselves inside the state vector.
This will be illustrated with a simple example: Say we have four states
x
=
[
(4.28)
y y
Y
and -y= ax 2 + by 2 , a complete square quadratic form
a 0 0 0
y = xT
0 0 0 0
0 0 b 0
(4.29)
x,
0 0 0 0
where only diagonal elements of B are non-zero. This leads us to write
76
i
=
2axi + 2byy,
which is also a complete square quadratic form
0 a
-Y= T
0 0
aO0O0O
a.
0 0 0 b
0
x
(4.30)
0 b 0
This example also illustrates how the BTB = [0] constraint holds. If the vehicle undergoes
noise excitation in each dimension at the acceleration level then
0 0
1 0
B=
(4.31)
,
0 0
0
1
and therefore
BB=
[0
0 0 01
0 0 0 0
(4.32)
Concerning Convolution
Even though Equation 4.10 only requires one-dimensional convolution, this still results in
O(N 2 ) computational complexity. Convolution can however be accomplished by computing
the characteristic functions of each of -yk and Atk and multiplying these (not convolving)
before applying the FII [52].
4At,.(r)
is determined analytically by Equation 4.18 already
and would require no additional computation.
fB (y)
however needs to be calculated within
[tO, tK-1] by applying Equation 4.11 at each time step. It can be shown that
characteristic function of
fB
#Pk(r)
(the
(y)) can be determined in a similar way to that described when
computing frk(y), but with an FFT algorithm [32]. The numerical complexity thus again
reduces to 0(Nlog 2 (N)).
4.2
Induced Error on Probability Calculations
We may differentiate among two different types of errors when calculating Pe(k).
The
first is approximation error, discussed in Section 4.2.1, where simplifying assumptions lead
77
us to calculate an approximate solution. The second induced error is that of numerical
calculation, discussed in Section 4.2.2, where we solve for the approximate solution to a
specific accuracy. Both these types of errors have very different origins and therefore require
very different methods of characterization.
4.2.1
Approximation Error
Approximation error may be further sub-divided into two major attributing factors. First
we find an upper bound approximation to Pc(k) (Equation 4.8) and second we assume a
measure of independence between -y and A-yk when performing this calculation (Equation 4.10), so our upper bound is itself calculated with an amount of error. We calculate a
signed calculation error because of the independence assumption, in order to narrow down
the envelope of error and more closely approximate the true value of Pc(k). Figure 4-3
illustrates the relationships between various collision probability calculations and their associated errors.
Two additional, and comparatively minor, sources of error that also require discussion
are those induced by quadratic approximation of the collision bound and also that of time
discretization. These issues are treated at the end of this section.
Upper bound Probability Error
Section 4.1.3 describes why the calculated Pc(k) = Peb(k) is an upper bound to the true
value denoted by P.r"e(k), with "ub" signifying "upper bound". It is infeasible to attempt
to solve directly for the amount of error between the calculation and the true value, since
the true value would need to be derived from an intractable calculation. We define the error
as
pube(k) = Peub(k) - PiT""(k) = Pc(k)
-
PCt**(k)
(4.33)
with "ube" signifying the "upper bound error".
Instead of venturing along the path of intractability, we rather create a lower bound
to PCrue(k) which we shall call Pib(k) <; Prue(k). Now use the lower bound to re-write
78
PC
PC. << I ----------- --------------------------.
«1C
truet
PC
th
Figure 4-3: Relationships Between PCre~t), PcLbt
t
ce"(t) and P b
Equation 4.33 as
pube (k)<; P(k) - pib(k).
(4.34)
A feasible lower bound would allow us to clamp the true probability of collision between
two calculable limits. One such intuitive and simple lower bound often used in estimation
theory and also employed as a measure of probability of collision by texts such as that by
Prandini et al[54] can be mathematically described as
pcb (k) = max[o,k]
1o fr (7)dy.
(4.35)
This is simply a matter of finding the largest probability of state space intersecting with
Df(t) within a period of time, as illustrated in Figure 4-4. Various texts exist where Equation 4.35 is used[12] [11] [2][44] [3] to approximate Pjrue(k). This is however a dangerous
course of action, since there is no guarantee on the lower bound's error, potentially compromising on P(SA) instead of on P(UA). Equation 4.35 is so simple however, that it only
adds complexity of O(nN) to the calculations performed at each time step. All being said,
the equation makes for a useful, empirically accurate and much-used lower bound, rather
than an estimate of Ptrue(k).
79
-
- ~
Lower Bound to Risk
P/b (t,
[
max |tet,,
fx
xx
.**Df (')
-----------
Hazard
W
-D5(
--------.
A------*---
tt
--
7 _State
Trajectory
fx(t)
fXy,)(x)
Figure 4-4: Calculating Pjb(t) Over a Finite Time Horizon Interval
Figure 4-3 illustrates the relationships between the different bounded approximations
and their errors and also includes P'r(t)to be discussed presently.
Calculation Error
Even calculation error because of an independence assumption is a complex problem. Independence between -y and
1. f
(Ay) = fA
A4
is equivalent to a combination of two assumptions:
(A ), thereby implying that the segmentation of probability space
inside and outside of Dj(t) and the subsequent creation of
frk (y)
has no effect on
the conditional distribution of Ay.
2. Convolution may be employed so that we may express
fr1(y)IB[k,kI
= frk(7) *
fA (A') IB[o,k-i],
(4.36)
where Equation 4.5 is the combination of items 1 and 2.
The two above enumerations will be referred to as Assumptions 1 and 2 respectively.
Before continuing, we acquire a number of mathematical tools describing the relationship between y and Ay in order to characterize error.
80
First we need to be able to describe expected values, variance and covariance of quadratic
forms. According to Jaschke [35][34], we may describe the expected value of a quadratic
form xTBx of the multi-variate normally distributed random vector x as
E(xTBx) = tr(BEx ) + pTBpx
(4.37)
where Ex and px are the covariance matrix and mean vector of x respectively. The variance
becomes
aTBx =
Var(xT Bx) = 2tr(BExBEx ) + 4IBExBpx
and the covariance between two quadratic forms yi = xTBx and Y2
(4.38)
= xTPx
can be de-
scribed by
PUxTBxoYxTPx
=
Cov(xT Bx, xTPx) = 2tr(BExPEx) + ApTB EPIpx.
(4.39)
From these equations we may now construct the mean vector of the joint density for h
=
[Yi, y2]T as
E[h] = [my1, my1]T
(4.40)
and the covariance matrix
Eih
=
~
(4.41)
PO[OY2
[PO'Y1 OrY2
0
J
Y2
where y1 = xTBx and Y2 = xTPX.
The next step is to be able to describe the orthogonal principal axes of the symmetric
matrix Eh. This is done through eigenvector decomposition and after some algebra the
slope of the principal axis with the same sign as p is given by
m
M
-[a2
+ O2+
(o4
-
2
20r2 O2 +
4
+
4p2o~,.2
(442
)
(4.42)
Y1
such that the correlation line yi = my2 + c is created. If we let
y
=
y1 and
Ai
=
Y2
then Figure 4-5 provides additional clarity with a contour plot of the joint density function
fr,(, A-i').
We will now describe the First-Order-Second-Moment (FOSM) change
81
2'
(y5 Aj')
ff,
L
1U
Am
MVol
mAt
Figure 4-5: Contour Plot of frp(7y,
in fAM (A)
because of an FOSM change in
frk
')
(y).
+-(MB - Mr)
m
with Correlation Line
This is the characterization of error
because of Assumption 1.
frk (y) and
fAtk
(Ay) are comprised of well-behaved summed and scaled versions of non-
central chi-square distributions. The maximum allowable Pe(k) within [to, th], denoted by
Pc.,
is a very small number (typically Pma_
< 0.05). Furthermore we make the reasonable
assumption that we take action along our flight path such that Pc(k) < Pc,.
always holds
(or equivalently we don't calculate Pe(k) > Pc,_. as described in Section 3.3.2). Using
these assumptions, we will calculate the expected influence on the mean and variance of
f) (A0) because of adjustments to frk(7), such as the removal of APcs probability space
Ark
to form frBk (y).
When we define
frk (Y)d7,
APck =
82
(4.43)
then, as before, we split the PDF of -y into
frk(,)
=
9Ak (,) + 9B(7)
=k
Akr
f=A
(7) +(1
(4.44)
B
- APck frB
Here the mean and variance of, for example, frk (-y) is denoted by mAk and 2A respeck.
tively. Now focus on the expected value of -y after the removal of probability space within
DI(tk), i.e., given that
7 > yo. This is expressed as
E[-y]|(7 > -yo) = E[-yk] = E[yk] + AE[yk],
(4.45)
where AE[7k] is the change in the mean of y after the cut. Notice that E[{y7]
therefore AE[yk]
E[-yk] and
0.
Equation 4.45 can be re-written as
AE[yk]
=
E[-ykJ - E[7k]
1-
k
(E[yk] - E[
Ak))
-APek7
AR~
_
1 - APC;
with E[
Aky]
= E[7]|(0
(mrk - MA)
= Amrk ,
(4.46)
; -y < yo). So if the mean of -y changes by AE[-yk], then according
to our assumptions we may employ Equation 4.42 to show that the mean of A- changes by
- AE[-yk]. With E[A - Bk] = E[AkJ + AE[Aik] we may then express
AmAP
AE[Atk]
2pporo22
-At.
(~~
2aor2
+ orr + 4p2 orC~
ARC,,
(mr
~1-P,
-mA,).
(4.47)
We describe the change in variance Aa,
2
2,
in a similar way such that
2
1
1 -APck
[Up,[Or
(r,)mr,+(m
2 APc
+
1-2APe rk
83
-APcmA
-APck
)2 -
m2
rk
APc
Ck
1-APck
2
10 A
r^
M - APckMAk
MAk
mrk -
MkmAk + (m-APCkk)
2 rk
-~
1 - APck
2
2
2
-
MAk]
-
ark
(4.48)
and can express the change in the variance of j as
Atk
)(ik)2Aa2
-
(4.49)
Equations 4.47 and 4.49 are representations of the changes in mean and variance of Ay,
given the first order approximation that the mapping of the first two moments from Y to
At may be accomplished through the correlation line with slope m. This approximation is
only feasible for small values of APck associated with small time steps of calculation At.
Notice that Equations 4.47 and 4.49 contain no reference to mBk or ars, but are expressed in terms of mAk and a.
This is the cause of much of the complexity of these
equations, but is crucial in order to determine upper bounds to errors at a later stage, since
frAk (y)
is strictly domain bounded within 0 < 17 < yo.
We now return to Assumption 2 and form the combined FOSM error estimates because
of Assumptions 1 and 2. Recall that the Assumption 2 error is induced by determining the
probability density of
'Yk+1 = Yk + Ak
(4.50)
through convolution of the conditional marginal PDF of -yk and the unconditional marginal
PDF Ak instead of operating on the joint density. Elementary statistics dictate that the
true mean and variance of 7k+1 be described by
*
mtrue
mrk+l = m " +kmtArk(N1
(4.51)
and
2true
ark+l
2true
ark
+
2true
ANrk
- 2pka OArk
true
(4.52)
respectively. The convolution process maintains Equation 4.51, so the only error in the
mean results from Assumption 1. Therefore, the error on mrk+1 accumulated within the
84
single time step [tk, tk+1] can be written as
Omrk+1
(4.53)
- "AmaN,
which is the direct application of Equation 4.46 such that
M ubrk+1
N
FOSM
mrk+1
calculated +mrk
=
(4.54)
The induced variance error over the same time step is described in a similar way, becoming
AO 2
012
2
-
Pkark 7Atk +
cAfNl),
(4.55)
with the first term originating from Equation 4.49. The second term, in keeping with our
FOSM approximation, is provided by Equation 4.39, because of Assumption 2. In this case
2-ub
2.FOSM
ok+1
rk+ 1
2.calculated +
.
-(4.5)k+1
(4.56)
In the spirit of FOSM analysis we can now perform an FOSM matching between
fu+1 (1)|B
::: fFOSM()|B[k k] and falculated( y)IB[k,] by creating a new random vari-
able through the transformation
o2-calculated+
rk+1
2
f7+ Ork+1
calculated
=
(4.57)
orfk+1
such that the FOSM approximation of
fub
(-y)|
becomes
calculated
frOSM(rk+1)|Bikk =
+calculated(g(rk+1))|B[k,kI.
f 2.calculated+
02
(4.58)
+1
rk+1
Vol k~l
The FOSM estimate of pub
Ck+1 then becomes
pFOSM(k
± 1) = pFOSM(k) +
j0
F'OSM( )B[kkd1-
85
pFOSM(k)]
(4.59)
so that the FOSM estimate probability error Perr then becomes
perr(k + 1) = pFOSM(k + 1) - Pc(k + 1),
(4.60)
where Pe(k + 1) is also equivalent to Pcalculated(k+ 1).
Note that PFOSM(k) may always be utilized as an FOSM corrected measure of Pc(k)
but then higher order moments need to be determined if we are interested in characterizing
the remaining error. A very simple way of applying this FOSM correction, with minimal
impact on the solution algorithm for Pc(k), is to adjust Yo at each time step to become
F,.calculated
'Yo
+
0calculated
rk+1
2
To -
Qmrk+l
(4.61)
Vehicle Applicability Tests
The preceding discussion focussed on finding PF"r(k), the accumulated FOSM error in the
calculation of an upper bound to collision risk by time t = tk. Such a real-time measure of
error is very useful when evaluating confidence in the upper bound calculation of Pc(k). It
would however be helpful to be able to determine a measure of a priori applicability of the
research described so far on a vehicle-obstacle problem, before implementing the bulk of the
propagation algorithm described so far. We now deal with the creation of a vehicle applicability test, whereby vehicle parameters, dynamics and process noise may be employed to
create a measure of the maximum expected Per"(k) within [to, th], i.e. maxvh<oo(Pe7). It
is then up to the developer to decide whether this measure of Pce"(k) is sufficiently small
when compared to
Pmax
and the alerting performance required for the application.
It should be made clear that we will not attempt to find the true max(Pc,") without
the calculation of Per"(k) described earlier in this chapter. In the interest of simplicity and
the creation of a basic heuristic test, we will quantify an upper bound to the dominant portion of Prr(k). Eventually, this approach boils down to calculating the number of standard
deviations between E[y] and the -yo boundary, calculating the maximum change in the number of standard deviations and then translating this into an amount of change in probability.
86
Inspection of Equations 4.47 to 4.49 and comparison with Equations 4.53 and 4.55 indicate that the so-called Assumption 2 in the previous Subsection is the dominant contributor
towards Prr(k). The dominance is asserted by the relatively small value of APc(k) 5 Pc.
and the amount of error is largely moderated by the covariance between -y and y, given by
Equation 4.39. Assumption 2 only contributes to the second term of Q,2
(Equation 4.55)
rk+1
so this is where we will focus our attention. As such, and only for this test, we assume that
a(4.62)
Qmrk+1
and
2
rk+1
-- 2Pkr
(4.63)
UAfk.
From Equation 4.2, the quadratic relationship between x and -y and also x and
y,
we
or and oAN are functions of the vehicle's closed loop dynamics and process
know that
noise. We will refer to the combination of these three aspects as the system model. For the
time being it is best to fix the system model to a representative form, to promote clarity
during the course of this explanation. For any one such system model, p is always maximized
at a point where the vehicle's velocity vector is aimed directly at the state origin, since y
and
i
are then perfectly aligned, i.e.
grad(-y) = C -grad(y),
(4.64)
where C is scalar and real-valued.
Next we perform a straight-forward state-propagation simulation (propagating closed
loop dynamics with a reference path and process noise in Simulink).
The simulation is
performed in the vicinity of the state origin, where the vehicle is made to fly directly at the
origin at a representative speed. This is the encounter where p is maximized. During the
simulation we also compute a2 ,
express
Q.2
and pkUrkospk using Equations 4.38 and 4.39. Now
as a ratio of o2 such that
rk
gr
Ra2
-=
k
87
rk
2
= -2p-rk
0
-> R,
P
=
F
rk
(4.65)
k
rk
where R, is the ratio of the standard deviation of the calculation error of Ywith respect to
the (unconditional) standard deviation of y.
Furthermore, as a worst-case (maximum calculation error), find the maximum value of
Rak and associated k = km where
ykm >
yo, and assume that all of Pc(km) = P,.
accumulated at k = kmn, meaning that Pc(km - 1) = 0 and Pc(K) = Pc(km).
is
R,, will
peak as the vehicle approaches the origin and will then change sign as the vehicle passes
through the origin. The shape of Rgk is dominated by the behavior of Pk and we denote the
maximum value of Rk as RUMAX. See Figure 4-6 for a typical illustration of the progression
of Ruk vs.
.
R(.0
+ S Fk
:;-
R
(yM
Fiue
-:
S
frth
imuaino
7 i
88
eil
UMAX
FlyigToadsth
rii
It is R,
Pm.a=
that allows us to express max( r
0.06, then Pc(km) = Pc,
). Let's look at an example: Say that
would be true when E[-y] is S standard deviations away
from yo, or mathematically
E[yk] = 7to + Sork,
(4.66)
where S is a positive real number that is dependent on the system model and derived
from fr
m(7).
For example, for the New Generation Mini II (NGM-II) UAV, described
in Section 5.1, S - 3 when
Pcma. =
0.02 and a bound on the standard deviation can be
expressed as
(3
-
Rek~ur
<
rue**
< (3 + ROak)cOrk
(4.67)
Since Rk = 0.15 for the NGM-II we can use frkm (7) to calculate that
maxvt<oo (Pc)| < 0.013
(4.68)
or max(Prr(k) ~ 0.22.
Pcmax
We may now iterate across various vehicle models in order to determine a worst-case
scenario. Also notice that the outcome of Equation 4.66 and the value
Ykm
in Figure 4-6
are seldom equal, but the worst case error results, as described in this section, when they
are equal.
At this point, it is worth pointing out that the vehicle applicability test is based on a
concept often encountered in reliability theory. The ratio
RUMA
is the induced deviation
of the so-called maximum safe MahalanobisDistance [5]. The deviation is caused purely by
calculation error.
4.2.2
Numerical Approximation
This subsection focusses on quantifying the induced calculation error when determining f(x)
through the FFT method described in Equations 4.19 to 4.25. Numerical approximation
of errors can be divided into three main contributing factors, namely round-off, sampling
and truncation error. Round-off error is usually negligible as long as sufficient numerical
precision is used to represent discrete values. Sampling error is a function of discretization
89
step size and is induced when sampled signals are integrated (or differentiated). Truncation
error is a function of the FFT method and is caused by the need to truncate the bounds
of the indefinite integral representing the FI integral. We will treat numerical error as a
sum of the latter two types of errors. (Aliasing error is negligible as long as we hold to the
frequency and sampling constraints described in Section 4.1.3).
Sampling and truncation errors are both functions of the interval between samples and
the number of samples taken. Therefore, as a practical matter, it is useful and important
to be able to express the sampling interval and number of samples required for a given
error in terms of the characteristic function. These errors are usually described in terms
of a probability density function [32] [17], but such a function is not available to us before
starting our calculation. Other approaches, like that of Bohman[9], are to experiment with
numerical parameters in order to obtain good convergence, clearly not a feasible approach
when calculation time and complexity and calculable risk error bounds are of importance. It
is however noteworthy that numerical approximation error may be reduced to an arbitrary
value at the cost of computational complexity.
Hughett [32] proved that the minimax (e =
eT
+ EN = Loo > 0) error of a two-sided
random variable x can be related to the minimum sampling period T and minimum number
of samples N of the density function fx (x) by bounding fx (x) and Ox (x) with exponential
functions. In this way we find that
(4.69)
T > 2(2A(2a - 1))1/a
ET(a -1)
and
N > 2 + 2(
2BT,3- 1
EN V
1
)1/(#-i)
(4.70)
VjxI > T/2
(4.71)
-
if
Ifx(x)|
Ajxj-
with a > 1 and
|Fx(u)|
and if
#
Blu|-# Viu\ > N/2T
(4.72)
> 1. For a given maximum allowable error E, Equations 4.69 and 4.70 may be
90
iterated for different values of E = ET + EN to find the minimum values of T and N. It is also
useful to note that if |fx(x)| = 0 Vxj
T/2 and |Fx(u)| = 0 Vjul
N/2T, then ET
=
0.
Hughett specifically derived Equations 4.69 to 4.72 since Equations 4.71 and 4.72 hold for
x expressed as a quadratic form of standard normal random variables. These bounds are
therefore directly applicable to our problem.
There is a subtle point to be observed when computing N and T for a given e: Unless x
is strongly skewed, it pays to perform the calculation of the IFFT to result in fx(x - xo),
where fx (x - xo) has approximately zero mean or median. This shift decreases the required
value of A to make Equation 4.71 hold. The shift is done by exploiting a property of the
FT, which is to multiply F(u) by e. 2
zo , where
xo is the amount of shift required to center
fx (x). After the calculation, fx (x) can be regained from fx (x - xo).
A difficulty does however arise in that A and a need to be determined from the shape
of the density function fx(x), not the characteristic function F(u), as promised. Recall
that we are attempting to calculate fx(x), so it is not yet available to substitute into
Equation 4.71. Hughett also shows that if we fix a = 2,q, with 7 being any non-negative
integer, then we may take
A = (-1)n2,(0),
where
#2,
(4.73)
is the 2q 'th derivative of the characteristic function, and Equation 4.71 will
always hold, as long as
#2,7(0)
actually exists. It pays to be able to increase a, since it
lowers the approximation error bound, as is clear from Equation 4.69.
4.2.3
Quadratic Approximation of Collision Bounds
A collision bound may be as simple as a spherical shell surrounding an aircraft, but may
also be any general quadratic shape described by Equation 4.2.
The contents of D 1 (t)
inside the collision bound is usually viewed as an unsafe domain around a vehicle. This
implies that the boundary is not "skin-tight", i.e. Df(t) includes space that is not taken
up by the vehicle itself (and possibly even vice versa). Such a representation is inevitable
when describing ^Yoin a quadratic fashion, since no relevant vehicles are truly ellipsoids
(airships probably come closest). Collision avoidance systems like TCAS are based on the
91
premise that vehicles are not to be allowed within a minimum distance of one-another, the
DMOD [67] [56]. For such systems, prediction of future transgression of this minimum miss
distance constitutes an unacceptably high risk of collision and warrants resolution action.
The quadratic collision metric presented in this thesis may be used to provide just such
a prediction of transgressing a minimum miss distance If5o. V7-o is however only a true
DMOD distance when Equation 4.2 is a purely spherical form. In this way, both TCAS
and the research documented in this dissertation really define collision risk as the risk of
entering domain of failure bubble around a vehicle.
It is worth pointing out that truer collision risk may be obtained (even though this is a
higher risk venture) by shaping the quadratic bound such that it more closely approximates
the shape of a vehicle. One such optimization may be suggested as
B*(t) = argminB(t)[j
VXE(D(t)nDf(t)C dx such that Dv(t)
c D1 (t)],
(4.74)
where D, (t) describes the volume contained within the vehicle and Df (t) describes the
usual y < yo. The time dependence is usually dropped, but it may sometimes be useful
to account for vehicles changing shape, such as the PCUAV system after docking [63] or
vehicles with variable wing sweep. This is left to the reader to implement if it is deemed
necessary for a specific problem.
4.2.4
Time Discretization Error
The numerical solution approach to finding Pc(t) described in this thesis requires time discretization in order to propagate vehicle state, state PDFs and the resulting accumulation
of Pc(t) within a finite time horizon window t = [to, th]. In discrete time the window is expressed at t = [to,
tK-1],
such that K discrete time samples exist between to and
The time step is defined as At
= tk+1 - tk
tK-1 = th-
and for this discussion is assumed to be constant
V k = [0, K - 1], where k is an integer.
When At -+ 0, the discretization most accurately represents the continuous time realm,
but at the cost of immense computational complexity. As with most discrete problems, we
need to design At in order to guarantee a measure of discrete representational accuracy. In
92
the next few paragraphs we will discuss two aspects that drive the size of At.
State and Collision Metric Propagation Accuracy
The solution algorithm to Pc(t) proposed in Figure 4-2 requires propagation of
f
(-y) in
order to find frk+1 (7y)
IB[k,k]. The propagation process employs convolution, an approximate
solution to the problem, to which the approximation error is discussed in Section 4.2.1.
During the propagation procedure however, we propagate the collision metric using
Yk+1
=
7k+
AYk
(4.75)
+
-AtYk.
=
This is a first order, or Euler, integration approach described in [53] and [72]. The dominant
second order error induced at each time step resulting from application of Euler integration
can be found from Taylor series expansion[22] to be
(4.76)
- = 1(At)2 k.
For the purposes of this discussion, we assume that the error in Equation 4.76 is representative of the total integration error. In addition, we assume that the vehicle in question is
commanded to fly a straight-line nominal trajectory at constant velocity V, at a unit circle
domain of failure and that the state covariance matrix Ex(t) is steady. In this case
therefore
ik
y=
x2
2
= 2xdz + 2xt . As a result, the integration error becomes
It= (At) 2 V 2.
(4.77)
The integration error is an absolute figure that needs to be compared with a relevant measure
of accuracy. As in previous sections, we employ the standard deviation
In short, we need to ask how large e7 may be when compared to
Rmax = max[oK
ark
ork as a reference.
Ork, i.e., find
(4.78)
RYa occurs when ark is minimized, since et is constant. For every individual application,
we need to find an acceptable Rm 2,, a process described in detail in Section 4.2.1's discus93
sion on vehicle applicability.
For the NGM-II vehicle example application in Chapter 5, when At = 0.002, then
RGnax = 0.002. The maximum resulting error in the calculation of Pc(th) because of discrete
time collision metric propagation therefore becomes about 0.8 percent. This is considered
to be negligible when compared to Pen(th), especially since the propagation error is an
upper bound.
Finally, recall that -y and y are calculated from propagation of the vehicle state x within
the finite horizon simulation time window. For the constant velocity case described above,
the state error is negligible for At = 0.002, in fact, it can be shown that there is always an
order of difference between the error incurred when propagating y and x respectively, with
pe"(th) being dominant. Therefore, if a given value of At provides a satisfactory value of
pe"(th), then there is no need to be concerned with state propagation error.
Risk Probability Accumulation Accuracy
The accumulation of Pc as a function of time is in itself an integration process. At every
discrete discretization time we calculate the added collision risk APe(k)
f70 gA
()dx[
-
Pc(k - 1)] and Pc(k) = Pc(k - 1) + APc(k) (Refer to Section 4.1.3). As is the practice with
such Euler integration processes, this integration error is approximated by the second order
term
pacc-err(t)
(At)2p(t)
=
2
The value of pacc-err(k) relative to Pc(k), needs to be sufficiently small in order to ensure
accurate discrete risk accumulation, or
max[0,.K-1]J
where (
pacc-err(k)
c Pck (k
480)
< 1. The maximum relative error in Equation 4.80 accumulates when the time rate
of change of APc(k) is maximized. This worst-case scenario again occurs when a vehicle is
94
on a straight-line heading towards the center of Df, at the maximum possible speed relative
to Df, and when Pe(k) is maximized. Since Pc(k)
Pcm., we should choose simulation
time step size such that
[APc(k) - APe(k - 1)] <
(4.81)
2c,.
This process requires selecting a value of At, using this At to complete a simulation of a
vehicle approaching perpendicular collision with Df, and iterating until Equation 4.81 holds
true. Note that the simulation does at least need to start with a small enough At to allow
a number of non-zero values of APc(k) to be calculated.
For the NGM-II for example, if we require ( = 0.01 and Pema.
=
0.05, then At = 0.001
results. Notice that for this example the value of At is very close to that calculated for
metric propagation accuracy. The smaller of the two values should be chosen.
The values of At suggested here are lower bounds in order to strictly ensure required
representational accuracy for all possible conflict scenarios. It is worth choosing At according to the measures described, but At should always be perturbed through the course
of a number of simulations in order to determine whether larger discretization steps yield
sufficiently accurate results. In this way we might considerably reduce computational effort.
It should also be realized that the head-on collision scenarios often used to characterize
worst-case discrete time performance seldom require the same amount of accuracy when
calculating Pc(t). In most head-on collision examples Pc(t) accumulates so quickly, and
collision is so clearly apparent, that accuracy under this special condition may be sacrificed.
The result would be an increase in At and therefore reduced computational complexity during most other conflict scenarios, without any visible loss of avoidance system performance.
4.3
4.3.1
Time and Resolution Issues
Finite Horizons
This research focusses on solving for collision risk within a finite horizon interval. Simulations of this nature within infinite horizons are folly because of the obvious infinite cal95
culation resources required. Such calculations might still be intractable for very large, yet
finite horizons. The intent is to create a method of solution that would be tractable within
a required simulation interval, and we therefore need to characterize both the tractability
or complexity of a solution and the size of the required time horizon, t,.
= th
- to. A re-
quired time horizon could be characterized in terms of a minimum, maximum or an optimal
duration.
Let us denote the upper and lower bounds on t, as
t"b
and
t respectively. The lower
bound is driven upwards only by the minimum amount of useful forward simulation time,
for example, the shortest amount of warning time required to avoid a hazard.
tb
is however
clamped down by both the maximum computationally tractable simulation time window,
denoted by te, and knowledge of vehicle intent.
Intent is possibly the most interesting of the influences on tw. It has been pointed out
(See Chapter 1) that collision avoidance systems for piloted vehicles suffer from an inability
to model pilot intent, and that this intent causes divergence in the extrapolation of state
estimation/prediction into the future. (See Figure 1-1). The intent of vehicles under autopilot control may be characterized more exactly, since such systems are usually guided
by known control laws and deterministic decision making processes. Even though this is
true, automated vehicles are still subject to events for which future intent is uncertain and
highly complex. Some of the processes might include unexpected human intervention, required rescheduling of flight paths because of mission changes, complex interactions between
numerous automated vehicles and/or system failure. These events make it impossible to
predict even the behavior of automated vehicles for an indeterminate period of time, thereby
capping tn,. We may however divide such events into two groups, those for which the onset
may be delayed by a known time
td
and those that cannot be delayed.
td
may, for example,
be the amount of time elapsing between input of a new navigation waypoint and the vehicle
acting on the waypoint change. The designer of a UAV may ideally be able to control this
time delay. As long as tw
< td
we are able to make use of finite horizon methods for collision
prediction, as stated in this thesis, without loss of generality, since no event of this nature
will occur within a given finite horizon simulation window. Additionally, the event is both
controlled and expected. Chief amongst the events that we will not attempt to compensate
96
for are immediate human intervention and unexpected systems failures.
It is up to the designer of a system to determine whether t, is upper bounded by
td
or by te, given the specifics of the system under development. The general tractability of
the algorithms developed here allows us to pose this problem as one of expense vs. requirements, where expense drives processing power and requirements drive the nature of a
vehicle's guidance and control solution. Figure 4-7 illustrates both the bounds of t,, and the
factors driving the bounds. On this figure, "Benefit" describes how useful a simulation time
window is, while "Complexity" describes the degree of computational complexity required
to solve for collision risk.
C
Q
0
LID
t
w
lb <
t wj <tub
- w
w-
d
tw
tCc
min(td,
t)
Figure 4-7: Factors Influencing the Size of te,
Furthermore, it is worth pointing out that the size of t.. has little effect on Pg"'(k), the
97
accumulated FOSM collision risk calculation error, discussed in Section 4.2. The calculation
errors are driven by the size of Pm,., the maximum allowable collision risk for a vehicle,
and since we assume that action is always taken to ensure that Pc(k) < Pc.
within any
tw, the error is both calculable and bounded within such an interval.
Interestingly enough, the upper bound of tw, given that it is either constrained by
td
or processing power, is also in a sense the optimum value of t,. The preceding discussion
shows that we always benefit by extending t,,, as long as we can afford the expense and
ensure that tw <
td.
A larger value of t. does provide more lead-time to collision event
prediction and more closely approximates an infinite horizon problem. The benefit does
however approach an asymptotic constant as t., increases.
The only remaining question is whether the upper bound of t" is useful, i.e., that the
time interval is long enough to provide adequate warning of a collision risk event. This is
an issue of adequate control authority and the ability to investigate risk along alternate
trajectories capable of avoiding typical collision scenarios. The lower bound of t,,, denoted
by tb, is captured from an avoidance scenario that requires immediate evasive action (at
t = to) where the vehicle avoids collision risk of Pc(th) = Pc(K - 1) > Pm.
This is
again dependent on the specific system under consideration. To clarify, a two-dimensional
example is provided:
Let Rsm(t) be the semi-major axis of an ellipsoid Df (t), and Rtr(t) be the minimum
turning radius of a vehicle, at maximum ground speed Vmax, where this turning radius is
employed as an avoidance trajectory (see Section 3.3). From simple geometry in Figure 4-8
we can show that
lb
KM
mI/(Rsm(t) + Rtr(t)) 2 - Rtr (t)2
~
a
Vmax
tw - M
(4.82)
where M is a real scalar multiplier with M > 1. M simply provides a safety margin, so
that Pc(h) < Pma.x
98
7
-
- =
__
Escape
Df (t)
Trajectory
RR,,,,(t)
JJ**
'Original
Trajectory
Rt, (tO)
Figure 4-8: 2D Example for Finding the Minimum Safe Propagation Time Window
4.3.2
Avoidance Trajectory Discretization Step Size
In Section 3.3 we describe how an infinite set of avoidance trajectories would need to be
created in order to form a continuous CSOC curve. We then discretized the time interval at
which avoidance trajectories are investigated, so as to create a finite number of compound
avoidance trajectories to examine. The relevant trajectory discretization step size was designated tAD. In this section we investigate the size of tAD and its influence on the avoidance
system. We do not attempt to create a general solution to the size of tAD, but rather discuss
the issues that should be considered during implementation on a specific platform.
The consideration is one of avoidance resolution vs. computational complexity. A high
"avoidance resolution" implies that we may investigate the delayed implementation of avoidance trajectories along very small increments of time. The maximum resolution is reached
when tAD =
At, the simulation time step size and the minimum resolution results when
tAD = t,,, the time horizon window size. The latter is equivalent to the traditional method
99
of solving for an SOC at t = to, and results in only one SOC, which is automatically the
trivial CSOC. On the other hand, having tAD =
At would result in unnecessary compu-
tational complexity. This smallest possible tAD may result in the ability to exploit the
smallest possible window of avoidance opportunity, and also allows us to wait until the
last possible opportunity before taking avoidance action, since the peak of P(SA) is more
precisely defined. In this way we reduce P(FA).
There is however little gain in decreasing tAD past the point where the resolution of the
resulting value of P(SA) is smaller than our ability to accurately calculate P(SA). Also,
intuitively there is very little gain in waiting until the last possible moment to apply an
avoidance maneuver when a collision is inevitable along the existing nominal trajectory N.
We will focus on the former as being in some sense the optimal value of tAD-
We recall that P(SA) = 1 - Pc(K - 1), where Pc(K - 1) is the probability of collision
accumulated within the duration of t. for a specific avoidance maneuver. We also recall
that the vehicle applicability test in Section 4.2.1 developed a heuristic measure of the
maximum calculation error incurred when finding Pc(K - 1). A developer may therefore
keep reducing tAD until
P(SA)ItAD+stAD
+ max(Prr(K - 1)) < P(SA)ItAD
(4.83)
for the CSOC across a large number of representative simulation runs.
Experience with simulation examples and general implementation of the collision avoidance strategy described in this thesis indicate that it is seldom worth reducing tAD to less
than
tAD
(4.84)
AAD,
where AAD is the vehicle's largest closed-loop position tracking time constant when commanded to follow a specific avoidance trajectory and . is a positive real number such that
usually 2 < , < 5, depending on the damping ration of AAD. For the NGM-II
a damping ratio (
a
K
a
3 with
0.6. A decrease in C would result in an increase in K. Equation 4.84
provides a good initial guess for tAD, which may then be optimized through iteration of
100
Equation 4.83, if desired.
4.3.3
Special forms
A number of special mathematical conditions exist which trivialize the calculation of the
PDF of a quadratic form of normally distributed random variables.
Special relationships between B and Ex
frk (7) will always be a chi-square distribution, more generally a non-centric chi-square
distribution, when any one of two equivalent sets of conditions hold [31]:
1. tr(B~xBEx) = tr(BEx), and mXBYxBmx
=
m3xBmx
2. BEx is idempotent, i.e. BEx = BExBEx
where Ex and mx are the covariance matrix and mean of x respectively. In such a case, the
non-centrality parameter of the distribution is A = 0.5mxBExBmx, degrees of freedom
f
= tr(BEx), mean mr =
f+
2
2A and variance of = 2f + 8A.
This special case allows us to create frk (-y) numerically from pre-compiled distribution
tables when the only non-zero entries in B are unit entries along its diagonal. At the same
time Ex would need to be an identity matrix. The advantages of this special property is twofold: Firstly we may somewhat reduce real-time computational complexity by transforming
B and Ex into the above-mentioned desired format, off-line; and Secondly this format is
useful for verifying and testing the implementation of the FI algorithm.
Special relationships between random variables
Another special case arises when all state variables are Independent Identically Distributed
(IID) normal random variables. The work of Fisher [23] and Cochran [14] show that when
sampling normal universes, the distribution and independence of quadratic forms are related
to their degrees of freedom. This result is usually referred to as Cochran's Theorem.
In more mathematical terms [39], if
x =
X1
...
101
xn
(4.85)
is a random sample from a random universe, with mean and variance denoted by yL and
respectively and y = xTBx. Denote a noncentral chi-square distribution with
a2
f degrees of
freedom and non-centrality parameter 62 by Xf(J2). Let A(1) > A2 > ... > A, be all the
distinct eigenvalues of B with multiplicities rl, r2, ..., rl respectively such that
i ri =n,
then
fr (y
Aix
=
2,(
),
(4.86)
i=1
where
A=
2(1rE,1n)/02
(4.87)
and
Ei =
(
(I - AjB), i = 1, ... l.
(4.88)
In this case ln denotes a nxl column vector of 1's.
This very useful form may be exploited if the state vector can be transformed into an
IID format. Such a transformation is not possible in general, but it is well worth exploring
this property for a specific application. Real-time computational savings would result at
the cost of calculating
4.4
xf(o)
tables off-line.
Chapter Summary
This chapter is developed from the following major assumptions:
P[x(t) E Df(t), exactly once in [tO,tk]]
~ P[x(t) E Df(t), at least once in [to, tk]]; the
boundary of a domain of failure may be expressed or approximated as a quadratic function
of state space; the quadratic collision metric, y, and
are assumed to be independent over
one discrete time step when calculating the amount of collision probability flow into Df (t)
(the error resulting from this assumption is characterized and can be corrected for); and,
when the collision avoidance system performs properly, there always exists a flight path
along which the probability of collision is less than some pre-defined maximum Pma., such
that Pcma
< 1.
An intuitive collision metric is introduced that is defined as a positive definite quadratic
measure of state space. In three-dimensional position space the metric manifests itself as a
102
spheroid domain of failure or hazard space surrounding vehicles and hazards. The quadratic
form is shown to act as a order reductive metric, compressing n-dimensional state space
into a one dimensional metric.
It is shown how the quadratic collision metric lends itself to solving the collision risk
algorithm illustrated in Figure 3-3 through simple and proven approximations, resulting in
the tractable algorithm described in Figure 4-2. The intractable O(N 2n) numerical complexity required by the solution in Figure 3-3 at each time step in a finite horizon window
of solution is thereby replaced by significantly reduced O(N 2 ) complexity.
We also illustrate how research lead by Jaschke[35] and Hughett[32] in the field of econometrics may be employed to efficiently solve for the quadratic PDFs required by the risk
solution algorithm in Figure 4-2. The calculations are shown to be solvable through Fourier
inversion of known probabilistic characteristic functions, and we show how to employ Inverse Fast Fourier Transforms (IFFT) in order to find the PDFs according to predefined
bounds of representation accuracy (error). The bounds are linked to the characteristic function, which is known exactly before inversion takes place. In this way, we may guarantee
and control calculation error bounds even before attempting IFFTs. The efficient IFFT
solutions of guaranteed accuracy allow further reduction of computational complexity at
each time step in the finite horizon to O(Nlog 2 (N)) for the algorithm in Figure 4-2.
A number of properties of the time derivative of the collision metric are derived. The
risk solution algorithm in Figure 4-2, for example, requires that the PDF of the time derivative of the quadratic collision metric be calculable. It is therefore shown that this derivative
is itself a quadratic form for most dynamic models of second order or more. It is also shown
that we may always convert the representation of the rate of change into a quadratic form
of second order or more. Jaschke and Hughett's methods may thus also be used to solve
for the required PDF of .
The calculation errors induced by the assumptions made to convert the intractable algorithm in Figure 3-3 to the tractable version displayed in Figure 4-2 are identified. Each
type of error is either quantified exactly, upper bounded or approximated through FOSM
103
methods, depending on the nature and severity of the error. These calculations may be performed during real-time simulation, in order to provide error estimates on each estimation
of collision risk.
A vehicle applicability test was devised which can be run off-line, before implementing
the bulk of the solution strategy. The test gauges the worst-case error of the solution strategy based on the bulk of the calculation error.
Practical aspects such as simulation step size, the time resolution of investigating avoidance trajectories and the horizon time length are investigated.
The major contributing
factors influencing these aspects are identified and a number of metrics are suggested in
order to constrain and optimize these quantities.
104
Chapter 5
Application Examples
This chapter focusses on application and analysis of the algorithm presented in Section 4.1,
for the determination of collision risk along a given vehicle trajectory. Yang and Kuchar[41] [42] [70] [69]
have shown how to employ instantaneous values and thresholds of P(SA) and P(UA) in an
alerting system. We will not attempt to redo these algorithms and applications provided in
their publications, but will rather focus on finding Pc(t) and analyzing its calculation error,
as described throughout Chapter 4. The reader is reminded of the relationships between
Pc(t) and P(SA) and P(UA) described by Equations 2.2 and 2.3.
Two simulated application examples are provided in the next two sections, with each
further sub-divided into two conflict scenarios. The first example is that of the PCUAV
NGM-II vehicle needing to avoid hazards such as tree trunks, while the second example
focusses on the calculation of collision risk between two large commercial transports. Both
applications are implemented in two dimensions, for the sake of simplicity and in order to
be able to distill meaningful conclusions from the simulation results. Results are analyzed
relative to the true probability of collision, Ptrue(th), at the end of each simulation horizon,
and percentage changes are described as Amount*f*
Mnge X
100.
In each simulation example, results are plotted only for part of the true finite horizon
time window t.,,. This is done in an effort to illustrate the phase of the simulation where risk
is accumulated. No significant change in Pc(t) is visible outside of t = [ti, t 2] in Figures 52, 5-3, 5-5 and 5-6, e.g., Pc(th) = Pc(t 2 ).
105
5.1
PCUAV NGM-I and a Stationary Hazard
5.1.1
Vehicle and Scenarios
Vehicle Description
The PCUAV NGM-II is an autonomous UAV developed within the MIT/Draper Technology Development Partnership Program (MDPP) at MIT[51][63].
The vehicle is part of
a cooperative group of UAVs known collectively as the PCUAV system and is operated
at medium and low altitudes. At these altitudes, surface obstacles such as trees (or tree
trunks), buildings, communication towers and the like are abundant.
The NGM-II is an 8kg, fixed-wing, single-engine aircraft in a pusher configuration, with
a wing-span of approximately 2m. It has two tail booms with a vertical stabilizer on each
boom and a horizontal stabilizer extending between the booms. The vehicle's length from
the tip of its nose to the trailing edge of its rudder is 1.5m. These dimensions and the
vehicle's handling capabilities combined with a cruising airspeed of around 20m/s, imply
that the NGM-II may be likened to a large sport or aerobatic model remote control (R/C)
aircraft.
The vehicle is known for its exceptionally accurate autonomous control system, able to
control its vertical and horizontal position to within 2m circular error probability (CEP)
(3-D) of a desired tracking input. Onboard sensors include a 5Hz Global Positioning System
(GPS), an Inertial Measurement Unit (IMU), pressure transducers for altitude and wind
speed measurement and differential GPS capability. The vehicle's control system is most
accurate when it locks onto a larger UAV within the PCUAV fleet, the Outboard Horizontal
Stabilizer (OHS) Parent vehicle, using an optical tracking system.
A simplified 2-dimensional, linear time-invariant (LTI), closed-loop dynamic model of
the aircraft, under free flight conditions, was created with the help of the architect of its
control system, Sanghyuk Park[51]. The closed loop vehicle dynamics are represented by
106
the fourth-order state space model
1
0
0
X1
-0.96
-0.8
0
0
fi1
X2
0
0
0
1
L2 .
0
0
-0.6
-1
X1
0
0
0
0.96
0
X2
0
0
J2
0
0.6
+
0
U(t) +
0
1 0
0 0
0
1
(5.1)
with
E[w(t)w(t)Tf =
The states are illustrated in Figure 5-1.
[
0.5
0
0
0.5
(5.2)
Conflict Scenarios
Two conflict scenarios where chosen. These scenarios represent typical, simple conflicts
where both the expected and worst-case accuracy of the proposed calculation algorithms
are probed, as illustrated in Figure 5-1.
The first scenario, on the right in Figure 5-1, is that of the NGM-II flying past a
stationary, tree trunk-sized hazard in close proximity, while commanded to hold a straightline trajectory with a forward airspeed of 20m/s, in a wind free environment. The simulation
starts with a mean state distribution of
mx = {2,0, -21.8,20]T
(5.3)
and state covariance matrix
EX
=
PX
0. 3255
0
0
0
0
0.3125
0
0
0
0
0.4167
0
0
0
0
0.25
=
(5.4)
at t = to. We will refer to this as the "fly-by" scenario. This is considered to be a typical scenario for the NGM-II, flying straight line trajectories between waypoints at cruise
speed. The proximity with which it passes the obstacle is representative of the maximum
107
X2
4
Fly-by
x1
Head-on
Approach
Figure 5-1: The Two PCUAV NGM-II Application Example Approach Scenarios
acceptable collision risk allowed for this vehicle, chosen to be Paa. = 0.05. The resulting
Pc(th) will therefore be representative of a level of risk when collision avoidance action will
typically be taken.
A mean hazard passage time (MPT) is provided for all fly-by scenarios, and is denoted
by
fMPT. The MPT indicates when a vehicle passed the center of the domain of failure.
More, precisely, for the straight-line flight paths and circular shapes of Df in this chapter,
tMPT
is the only time when the mean of
f is zero, i.e., it is changing sign. In this instance,
fMPT = 1.09 seconds.
The second scenario, on the left in Figure 5-1, has the NGM-II approach a stationary,
tree trunk-sized hazard on a direct collision course for the center of the hazard.
108
The
simulation starts with a mean state distribution of
mx = [0, 0, -24,
(5.5)
2 01T
and state covariance matrix
0.3255
EX = P
=
0
0
0
0
0
0.3125
0.4167
0
0
0
0
0
0
0
(5.6)
0.25
at t = to. This will be referred to as the "head-on" scenario. Such a conflict represents the
worst-case (most calculation error, see Section 4.2.1) performance of the proposed collision
probability calculation algorithm. This scenario also represents a clear case where collision
is inevitable and where the probability of collision accumulates most rapidly over time. We
will make use of the "head-on" simulation in order to validate the vehicle applicability test
described in Section 4.2.1, and therefore terminate the simulation when Pc(t) ~ P..
Notice that no distinction is made between nominal and avoidance trajectories, since the
algorithm itself makes no real distinction. It can however be proposed that the first scenario
may be the result of either a nominal trajectory, or for example, an avoidance trajectory
from a nominal "head-on" trajectory. In both cases, the domain of failure (hazard) Df
is described as a circular space around the vehicle, with unit radius (in meters). This is
considered to correspond to an appropriate value of -yo, given the vehicle's dimensions and
a tree trunk of approximately 15cm radius.
5.1.2
Results
Fly-by Simulation
Figure 5-2 illustrates the results of accumulation of risk within a time window t = [ti, t 2],
where to < ti < t 2 < th, within a certain finite horizon time window. In each instance
throughout this chapter, ti = to + 1. The four distinct plots on this figure are related to
each other as follows.
109
0o05
0-045
-
0-04
--
0035 -c
003 -0.0
-
0 0050
pr-(
-
t
0
1
1.051
Time in seconds
Figure 5-2: Risk Accumulation for PCUAV NGM-II Hazard Fly-by
The solid line represents a close approximation of the true probability of collision at time
t
=
th. This is denoted by Ptrue(t), or the "True" probability of collision, introduced in
Section 4.1.3. This close approximate is calculated from an exhaustive and time-consuming,
model-predictive Monte-Carlo simulation and is considered to be the benchmark for comparing the performance of the algorithm developed in Chapter 4.
The algorithms for calculation of collision risk, developed in this dissertation, are represented by two plots in Figure 5-2. One is denoted by Pc(t), the calculated upper bound to
collision risk, and another is denoted by Pfosm(t). The latter is an FOSM error corrected
value of Pc(t), accounting for calculation error induced by the independence assumption
discussed in Section 4.2.1.
A lower bound probability of collision, Pb(t) is determined, as described by Equation 4.35. This representation of probability of collision is suggested by Prandini et al[54]
and also by number of other publications on collision avoidance[12] [11] [2] [44] [3].
110
Adding
this plot to the figure allows us to clamp Pj'"(t) between upper and lower bounds. In
addition, it allows us to compare our work with comparable recent research on probabilistic
collision avoidance.
Figure 5-2 illustrates how Pc(t) and PfOSM (t) are accumulated over time, as the vehicle
passes the hazard, again emphasizing the similarities between collision probability and the
first passage time problem discussed in Section 2.3.2. The figure shows the upper-bounded
nature of Pc(th) = Pc(t 2 ), since it is larger than PcrUe(t) and also shows the lower bounded
nature of Pib(t), which is clearly smaller than Pcr"e(t)
PfSM(th) = Pc'OSM(t 2 ) is within eight percent of Pc""(t) and Pc(th) is within six per-
cent of Pcr"e(th). The seemingly more accurate Pc(t) is not evidence of failure of the FOSM
error correction method. Pc(t) is an approximation of the upper bound of the probability
of collision, denoted by Pcb(t), and PFOSM(th) is a more exact approximate of Pcb(t). It is
circumstantial that Pe(th) < PFOSM(th), and it can be shown that the opposite inequality
holds under different simulation conditions. Specifically, when the finite horizon accumulates more probability of collision while -y and
A
are negatively correlated (p < 0), then
Pc(th) < PFOSM(th), Pc(th) > PcOSM(th) when more risk is accumulated while p > 0.
This is obvious when assessing Equation 4.55.
The independence assumption labelled "Assumption 1" in Section 4.2.1 accounts for the
approximately two percent difference between P!OSM(th) and Pc(th). The figure is well
below the worst-case upper bound error of twenty-two percent derived during the vehicle
applicability test example in Section 4.2.1. This can be expected, since the simulation is not
a worst-case scenario, such as the head-on approach. It is under this worst-case assumption
that the bound was derived.
pib(t) is of special interest in Figure 5-2, since it is approximately fifteen percent smaller
than Pjrue(th). Using the lower bound alone in order to obtain an estimate of PrueTh)
would be much simpler to accomplish than the algorithm used to find Pe(th). The danger
involved in trusting in a lower bound of collision risk, combined with the associated estimation error, would however make this a risky venture. The difference between Plb(t) and
111
Pc(th) may however serve as an insightful method of bounding Ptrue(th).
It is interesting to note that Pjb(t) quickly becomes a progressively worse measure of
Pjrue(th), the more time is spent in the vicinity of D 1 . One such example is when orbiting
a circular Df at a constant radius. Pib(t) will not grow as time progresses, while both
Ptr"e(th) and Pc(th) would increase.
Head-on Simulation
The head-on approach shown in Figure 5-1 incites the worst-case difference between pFOSM(th)
and Pc(th) because of the maximal correlation between y and j when these vectors have colinear gradients (see Equation 4.64 and the surrounding discussion). Figure 5-3 illustrates
the results of the simulation.
Note the exponential increase in Pc(th) and PFOSM(th) over time, as the vehicle inevitably enters Df. The increase in risk is sudden and evolves from almost zero risk to
pFOSM(th) = 0.053 over the course of only about 0.06 seconds. It is this sudden increase in
risk that accounts for the minimal difference between Pcj(th) and ptrue(th). There is very
little probability flow out of Df, with the commanded cruise speed of 20m/s dominating
the flow into Df. This is therefore a scenario where the vehicle will spend very little time
in close vicinity to Df, resulting in an accurate Pcb(th), approximately four percent smaller
than Ptrue(th)
At the same time, pFOSM(th) is also accurate, being approximately eight percent larger
than
Prue(th). PEOSM (th)
is therefore of similar accuracy for both the "fly-by" and "head-
on" simulations. Pc(th) undershoots
Prue(th) by
about 10 percent.
PfOSM (th) is roughly eighteen percent larger than Pc(th), which is close to the 22
percent approximate worst case "Assumption 1" calculation error predicted in Section 4.2.1.
This simulation (and many similar simulations) therefore validates the approximate vehicle
applicability test described in Section 4.2.1.
112
0.06
0.05
M
pFOS ~
P
.0
M
0.04-
P
~CJ
----
W
PC"
W
Zcc
M
0-020.01r
1
1.1
1.05
Time in seconds
Figure 5-3: Risk Accumulation for PCUAV NGM-II Head-on Hazard Approach
5.2
5.2.1
Two Large Commercial Transports
Vehicle and Scenarios
Vehicle Description
This application example aims to represent conflict scenarios between two large commercial
transports such as the Boeing 747-400, or vehicles of equivalent size and dynamics, under
autopilot control.
The simplified two-dimensional LTI, closed-loop model of the vehicle, under autopilot
control, was developed with the aid of Tom Reynolds at the International Center for Air
Transportation (ICAT) at MIT and according to dynamics provided by Roskam[57]. A
113
fourth order representation can be summarized as
0
0
0.2
0
X1
0
1
0
0
X1
21
-0.2
-1
0
0
±1
X2
0
0
0
1
X2
0
0
-2
0
0
-0.2
-1
±2
0
0.2
+
0 0
1 0
U(t)+
0
0
0
1
w(t); (5.7)
with
E[w(t)w(t)T ] =
0.5
0.
0
0.5
(5.8)
The model is further transformed to vehicle-relative space as to adhere to the singlevehicle-single-obstacle problem definition described in Section 3.2. The transformation results in a single vehicle model in conflict with a stationary Df. Df is described by a circular
region with 40m radius, roughly accounting for the dimensions of two large transports.
Conflict Scenarios
Conflict scenarios similar to that described for the PCUAV NGM-II vehicle are investigated.
In the case of the two-vehicle transport collision problem however, the fly-by scenario, on
the right in Figure. 5-4, corresponds roughly to two vehicles crossing each other's flight
trajectories at the same altitude. The fly-by simulation starts with a mean state distribution
of
mx = [43, 0, -248,
20 0
(5.9)
]T
and state covariance matrix
Ex = Px =
2.5
0
0
0
0
0.5
0
0
0
0
2.5
0
0
0
0
0.5
(5.10)
at t = to. For this scenario, the MPT is calculated as
fMPT =
1.24 seconds.
The head-on conflict scenario, on the left in Figure 5-4, corresponds directly to two vehicles artificially commanded along trajectories that would result in direct head-on collision.
114
The simulation starts with a mean state distribution of
mx = [0, 0, -246.5,
2 0 0 1T
(5.11)
and state covariance matrix
Ex = Px=
2.5
0
0
0
0
0
0.5
0
0
2.5
0
0
0
0
0
0.5
(5.12)
at t = to.
Both scenarios are typical of situations that might be encountered under straight-line autopilot control between navigation waypoints. In this case however, the head-on simulation
is prolonged in order to accumulate very large risk, about five times that of Pm_. = 0.05
chosen for these aircraft. Recall that we usually avoid calculating Pc(t) > Pc.,
since a
number of analyses in Chapter 4 depend on Pc(t) being small, including the vehicle applicability test and the FOSM calculation error corrections. The increased risk is therefore
intended to expose inclement influences on the calculation of PFOSM(t) and the vehicle
application test, if any.
5.2.2
Results
Fly-by Simulation
The fly-by simulation results in Figure 5-5 again attest to the accuracy of Pc(th) and
PfOSM (th)
when compared to Ptrue(th). P 0 (th) is approximately 8 percent larger than
Pjrue(th) and PfOSM(th) is about ten percent larger.
The difference between Pe(th) and PFOSM(th) is only about 2 percent, much smaller
than the predicted 18 percent approximate maximum from a vehicle applicability test. We
again expected the improved calculation error performance, since the maximal error (as
usual) was derived from a worst-case head-on approach.
115
.1-
-,a.
a
,-
a
.
I&
.. .
_w
-
.
m
a
a
X2
A
| l-X
-Fly-by
at
Head-on
Approach
f",~)
Figure 5-4: The Two Transport vs. Transport Application Example Approach Scenarios
116
4
-
-q
.
W
-
T.
..
-
-
0.04
0.035
P
-
0.03
Ci
PFOSM(t)
--
CC
0 25 -
-.
P
-
0.02 S 0.015 -0.01 0.005 0
.0-1
1.25
1.2
1.15
Time in seconds
1.3
1.35
1.4
Figure 5-5: Risk Accumulation for Transport Fly-by
Pcb(th) is approximately 18 percent smaller than Pct"e(th). Approximating Ptrue(th)
with pib (th) would grow progressively worse with increased durations of exposure to Df.
Further simulations have illustrated this decreased accuracy when the vehicles' relative velocities are reduced.
Typical risk accumulation is evident as a function of exposure time and proximity to
Df, similar to the results from other simulations, including the one resulting in Figure 5-2.
Head-on Simulation
The head-on conflict scenario for two large commercial transports was simulated and the
collision probability results, for part of one finite time horizon, are shown in Figure 5-6.
This scenario was specifically tailored to again induce the worst case performance of the
algorithm for Pc(th).
The typical exponential increase in risk probability is evident as the two vehicles in-
117
0.35
03-
P (t)
--..
025F
C
0
U,
--
0
pCFOSM \
Pr"'(t
0.2-
0
0
-o
w
'P--
-D
0
0~
0.1-
o0ak
0
1
1.005
1.01
1.02
1.015
1.025
1.03
1.035
Time in seconds
Figure 5-6: Risk Accumulation for Transport Head-on Approach
evitably head towards a collision. Relative vehicle velocities are approximately 200m/s for
this simulation, resulting in a rapid rise in collision risk over time. As a consequence, Pb(th)
and
Pjrue(th)
are separated by about a percentage point. PfoSM(th) is approximately 4
percent larger than
Ptrue(th).
Although the lower bound is therefore more accurate during
this scenario, PFOSM(th) is very accurate.
Pc(th) is 28 percent smaller than PFOSM(th), only 4 percent larger than the 24 percent
value calculated with the approximate vehicle applicability test for this worst case scenario.
When PFOSM(t) = 0.05, Pc(th) is only 22 percent smaller. As PFOSM(th) increases, the
vehicle applicability test does understandably become slightly less accurate, since the test
hinges on the assumption that pFOSM(th) < Pc.
is small. At the same time, and for the
same reason, when PFOSM(t) = 0.05, it is only 2 percent larger than Ptrue(t), implying
that PEOSM (t) is itself more accurate at lower collision risk. In all though, this simulation
again validates the vehicle applicability test developed in Chapter 4 and PfOSM (t) again
provides an accurate estimate of collision risk, with higher performance when pFOSM (th)
118
is small.
5.3
Chapter Summary
The risk calculation algorithm in Figure 4-2 is implemented in a Matlab and Simulink
environment. Two application examples are considered: A small UAV, the PCUAV NGMII, encountering a tree trunk sized hazard; and two large commercial transports in hazardous
proximity to each other. Two conflict scenarios are tested, illustrated and discussed for each
application. It is shown that both the risk calculation and the error approximations perform
expectedly well, thereby validating the developed approach.
119
120
INIMININK,
A61k._ _'_ ;-
-
1-
Chapter 6
Conclusions and Future Work
This thesis further develops the field of probabilistic collision avoidance systems, specifically
for autonomous aircraft, and builds primarily upon the works of Yang and Kuchar[70] [69] [42] [41],
but also on approaches by Paielli and Erzberger [49] [50] and Prandini et al[54]. The problem
is treated in the form of a multi-dimensional, non-cooperative game between two vehicles,
or equivalently, one vehicle and an obstacle.
6.1
Primary Contributions
The following is a list of the primary contributions of this research:
1. Order reductive collision metrics are applied in order to provide a tractable algorithmic
solution to the normally intractable first passage time problem of multi-dimensional
collision risk calculation within a finite time horizon.
2. A tractable real-time algorithm is developed for the calculation of collision risk as a
function of time along any deterministic flight trajectory.
3. Calculation and approximation errors resulting from application of the algorithm are
characterized and quantified.
4. Simulations are employed to verify the accuracy of the collision risk algorithm and
major calculation error components and to probe the applicability of the research to
various application examples and conflict scenarios.
121
5. The concept of a system operating characteristic (SOC), developed by Kuchar[41], in
aid of calculating alerting thresholds, is expanded to account for the safety benefits
of delaying collision avoidance action. A condensed SOC (CSOC) results.
The remainder of this chapter describes the applicability of the research to various
collision avoidance problems, further summarizes the conclusions and contributions made
during the course of the research and lists future research opportunities stemming from this
work.
6.2
Applicability and Comparisons
Section 2.2.3 describes how most conflict alerting systems may be divided into three categories, namely: Single path, worst case and probabilistic methods of approach. Single path
and worst case approaches are typically attempts at simplifying the generally intractable
probabilistic approach, in order to provide a tractable solution. Such approaches are therefore usually applicable to specialized collision avoidance problems and many are of an ad
hoc nature. The method developed in this thesis, follows the probabilistic approach. This
approach typically gains generality of application and accuracy, at the cost of calculation
complexity. We show how calculation complexity may be reduced, while still retaining most
of the advantages of the probabilistic approach.
The work expands upon the probabilistic approaches followed by Kuchar, Yang, Paielli
and Erzberger and Prandini et al.
It is closest in comparison to that of Paielli and
Erzberger's approach, since their work is also only applicable to the two-vehicle problem
and follows a similar stochastic development. Kuchar and Yang's work is applicable to multiple domains of failure, but their Monte Carlo approach sets it apart, and their solution to
collision risk can be more approximate, in order to facilitate real-time tractability.
Our work primarily differs from that of Paiell and Erzberger in the following ways:
1. We calculate an upper bound to collision risk, which closely approximates the true
collision risk, while Paielli and Erzberger's results may return upper or lower bound
estimates, depending on the amount of time spent in proximity to a hazard. A feasible
122
upper bound provides added robustness to collision avoidance, while a lower bound
may compromise on safety.
2. We developed an approach which is of more general application to collision avoidance
problems. This fact stems from the fact that they assume constant vehicle velocities
and constant state distributions over the conflict interval, while also not recognizing
that additional conflict risk arises from exposure to a hazard over time. These assumptions may become problematic when considering, for example, a UAV loitering
in the vicinity of a hazard, accumulating risk over time. In general, our approach is
therefore also more accurate.
3. Our approach, even though it is tractable in real-time, is of much higher computational
complexity than that of Paielli and Erzberger. Increased complexity might prohibit
use on small, uncomplicated and inexpensive vehicles with limited processing power.
The work developed in this dissertation is aimed at application to autonomous vehicles.
In cases where human intent may be modelled stochastically, it may also be applied to
human piloted vehicles, with an associated increase in false alarm rates and a reduction in
correct detection rates. The reduction in performance would result from a typical increase
in uncertainty of intent, not algorithmic error.
Furthermore, the work is applicable to two-vehicle fixed and rotary wing aircraft problems, during glancing and loitering encounters with hazards, making it of very general use.
The work is however not applicable to scenarios where the precise and complex shapes of
hazards need to be maintained during risk calculation, such as the PCUAV autonomous
mid-air docking system[51}.
6.3
Summary and Conclusions
The following is a more detailed list of the approach and contributions of this research:
1. It is illustrated that automatic collision avoidance systems for autonomous vehicles
need to be developed in order to: Allow UAVs to operate in FAA controlled airspace;
increase the effectiveness of commercial aircraft collision avoidance when under au123
topilot control; and ensure safety when piloted and autonomous vehicles are required
to interact.
2. Differences between piloted and autonomous vehicles that influence conflict avoidance
design are discussed. The effectiveness of alerting systems for autonomous vehicles
is shown to be less sensitive to the occurrence of false alarms.
Uncertainties usu-
ally associated with future pilot intent can be reduced dramatically by deterministic
autopilots, thereby opening the door to more precise probabilistic conflict detection.
3. The autonomous conflict detection and collision avoidance problem is cast into a state
space problem, similar to that developed by Kuchar[41].
The autonomous vehicle
problem is then extended into a closed loop representation where it is shown that
mean and covariance propagation may be used to estimate future state uncertainty
through finite horizon model predictive simulation.
This development is a direct
consequence of the reduction of uncertainty of pilot intent because of autonomous
control. Pailli and Erzberger[50] also illustrate a similar initial development, but for
vehicles with constant velocity.
4. The calculation of collision risk within a finite time horizon is shown to be equivalent to problems encountered in a number of separate fields of research. Two such
fields are: Reliability theory, especially as applied to civil engineering problems where
catastrophic failures are characterized as a function of time; and the first passage
time problem described by Kolmogorov equations, or as special cases thereof, such
as the Fokker-Planck equation often encountered in physical problems and stochastic
control.
5. The calculation of collision risk within a finite and discretized time horizon is cast
as a reliability theory problem and developed from first principles. The solution is
developed as a set of discrete propagation equations illustrated in Figure 3-3 and it is
shown that the calculations are numerically intractable when real-time solutions are
required.
6. The notion of the system operating characteristic (SOC) [42] [41] employed as a tradeoff between P(SA) and P(UA) when making alerting decisions is extended. The SOC
at any point in time is based on the outcome of making an avoidance decision at that
124
time. It is shown that the effects of time-delayed avoidance action may be incorporated into the SOC representation, resulting in a best-case or time condensed SOC
(CSOC), for a specific time horizon and a given set avoidance trajectories. This allows us to make use of signal detection theory in order to optimize alerting thresholds,
while at the same time investigating the pay-off of delayed avoidance action. As a
direct consequence, the work of Kuchar may be employed when optimizing alerting
thresholds[411.
7. An intuitive collision metric is introduced that is defined as a positive definite quadratic
measure of state space. In three-dimensional position space the metric manifests itself
as a spheroid domain of failure or hazard space surrounding vehicles and hazards. The
quadratic form is shown to act as a order reductive metric, compressing n-dimensional
state space into a one dimensional metric.
8. It is shown how the quadratic collision metric lends itself to solving the collision
risk algorithm illustrated in Figure 3-3 through simple and proven approximations,
resulting in the tractable algorithm described in Figure 4-2. The intractable O(N 2n)
numerical complexity required by the solution in Figure 3-3 at each time step in a
finite horizon window of solution is thereby replaced by significantly reduced O(N 2)
complexity.
9. We also illustrate how research lead by Jaschke[35] and Hughett[32] in the field of
econometrics may be employed to efficiently solve for the quadratic PDFs required by
the risk solution algorithm in Figure 4-2. The calculations are shown to be solvable
through Fourier inversion of known probabilistic characteristic functions, and we show
how to employ Inverse Fast Fourier Transforms (IFFT) in order to find the PDFs
according to predefined bounds of representation accuracy (error). The bounds are
linked to the characteristic function, which is known exactly before inversion takes
place.
In this way, we may guarantee and control calculation error bounds even
before attempting IFFTs. The efficient IFFT solutions of guaranteed accuracy allow
further reduction of computational complexity at each time step in the finite horizon
to O(Nlog 2 (N)) for the algorithm in Figure 4-2.
10. A number of properties of the time derivative of the collision metric are derived. The
125
risk solution algorithm in Figure 4-2, for example, requires that the PDF of the time
derivative of the quadratic collision metric be calculable. It is therefore shown that
this derivative is itself a quadratic form for most dynamic models of second order or
more. It is also shown that we may always convert the representation of the rate of
change into a quadratic form of second order or more. Jaschke and Hughett's methods
may thus also be used to solve for the required PDF of y.
11. The calculation errors induced by the assumptions made to convert the intractable
algorithm in Figure 3-3 to the tractable version displayed in Figure 4-2 are identified. Each type of error is either quantified exactly, upper bounded or approximated
through FOSM methods, depending on the nature and severity of the error. These
calculations may be performed during real-time simulation, in order to provide error
estimates on each estimation of collision risk.
12. A vehicle applicability test was devised which can be run off-line, before implementing
the bulk of the solution strategy. The test gauges the worst-case error of the solution
strategy based on the bulk of the calculation error.
13. Practical aspects such as simulation step size, the time resolution of investigating
avoidance trajectories and the horizon time length are investigated. The major contributing factors influencing these aspects are identified and a number of metrics are
suggested in order to constrain and optimize these quantities.
14. The risk calculation algorithm in Figure 4-2 is implemented in a Matlab and Simulink
environment. Two application examples are considered: A small UAV, the PCUAV
NGM-II, encountering a tree trunk sized hazard; and two large commercial transports
in hazardous proximity to each other. Two conflict scenarios are tested, illustrated
and discussed for each application. It is shown that both the risk calculation and
the error approximations perform expectedly well, thereby validating the developed
approach. At the same time, results are compared to that of a lower bound estimate
of collision risk suggested by Prandini et al and a host of other publications.
126
6.4
Future Work
Various avenues of future work result from the research detailed in this thesis, many of which
fall outside of the classical realm of probabilistic collision avoidance. A select number of
avenues are briefly discussed in this section.
6.4.1
General State Avoidance
The problem of collision avoidance is generally treated as one of avoiding a time variant
positional state space, i.e., Df(t) C R' where m < 3. That is, the problem is a subset of
the larger problem of general state avoidance, where m < n.
General state avoidance may be employed in a wide variety of situations, including:
Avoiding hazardous domains of position and velocity during autonomous landing; Avoiding
hazardous domains position, velocity and acceleration for agile maneuvering autonomous
vehicles; and Avoiding hazardous domains of orientation outside of, for example, GPS antenna beam width and tilt sensor range.
The research detailed in this document may be adapted for use as a general state avoidance system, or at least for determining the risk of transgression into a general hazardous
state domain. In order to do this, either the D1 (t) of Dc (t) needs to be quadratically
bounded or approximated as such. Most of the derivations in Chapter 4 may be extended
to general quadratic metrics of state, not only operating on position space. The metric
must however always be positive definite, and the FOSM error correction of Pc(t) would
need to be extended to be able to deal with quadratic forms that are not complete squares.
6.4.2
Control and Trajectory Scheduling for Minimal Risk
The probabilistic methods discussed here are aimed at the evaluation of conflict risk along a
pre-defined set of flight trajectories. The size and variety of this set and the time-resolution
at which avoidance trajectories are investigated can be traded off against effectiveness of
collision avoidance P(SA) and computational complexity. At least three schools of thought
exist that allow us to revise the notion of employing pre-defined sets of trajectories:
127
Firstly, the set of trajectories may be very large, but a selection algorithm may be developed to rank or prune the set, in real-time, given the current state estimation and the
nominal trajectory. In this way, only a selected subset of the original set, and of tractable
size, may be entertained during each finite horizon.
Secondly, the concept of pre-defined trajectories may be replaced entirely by alternative
forms of collision avoidance control. The model-predictive nature and step-wise propagation
embedded in the approach detailed in this thesis, allow for the application of optimization
methods employing, for example, scaled control input primitives along the course of a finite
horizon in order to optimize the collision risk outcome. In this way, collision avoidance is a
matter of tweaking nominal trajectories to bend around hazards, without radical changes
in trajectory. Similar research is being undertaken at the Charles Stark Draper Laboratory.
Thirdly, pre-defined sets of avoidance trajectories may be replaced by a single nominal
path, with control around this path being altered by non-linear control laws. Such control
laws may be dormant when collision risk is minimal, but may become effective in close
proximity to hazards. These laws may either be gain scheduled according to collision risk,
or may be more directly reliant on geometrical properties of conflict scenarios such as laws of
inverse proportional navigation. Methods of adjoint analysis may, for example, be applied
in order to efficiently calculate miss distances and trajectory changes.
6.4.3
n-Multiple and Compound Vehicles
The work may be extended from non-cooperative two vehicle or one-vehicle-one-obstacle applications. Direct application of the research would imply that n-multiple vehicle problems
be divided into (n -1) separate conflict analyses running on each one of n different vehicles.
This might result in extreme computational burden. Complicated scenarios may also be
envisioned around airports or within fleets of cooperative UAVs, where non-cooperation
might lead to acutely sub-optimal avoidance capability. Various methods of cooperative
data analysis should be researched if this work is to be adapted to such scenarios.
Finally, each vehicle might also be divided into compound sets of intersecting, or nonintersecting domains of failure. This is linked to the computational complexity we see when
128
dealing with n-multiple vehicles, and intersecting domains would pose yet another probabilistic challenge.
Dividing the multi-vehicle collision risk problem into 2-vehicle pairs, however also begs
recognition of the fact that the risk of collision between two vehicles is conditional on the
risk between all other pairs. This issue would have to be resolved before the research can
be applied to n-multiple and compound vehicles.
129
130
- -----------
M WA .0iffimmia-
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138
Appendix A
Matlab Simulation Code
A.1
Sample Code for Calculation of Probability of Collision
Within One Finite Horizon
% Collision Risk Calculation Algorithm Main File
% Thomas Jones
% Units:
%
All times in seconds
%
All translations in m
%
All velocities in m/s
%
INCLUDING FOSM ERROR CORRECTION
X This example calculates collision risk for a short, specific
% PCUAV NGM-II fly-by encounter, longer intervals
% require discretization step size scheduling not shown here
close all;
clear all;
Xfigure;
% SETUP
139
% Starttime in seconds
t0=0;
% Total simulation time in seconds
totalsimtime=0.11;
% Lateral control time constant in rad/s
vehicletimeconstant=1.5;
% Collision zone radius in meters
RO=1;
GamO=RO^2;
% Deffinition of collision zone
Gam=[1 0 0 0;0 0 0 0;0 0 1 0;0 0 0 0];
% Simplified collision zone rate for error calculation
GamD=[O 1 0 0;1 0 0 0;0 0 0 1;0 0 1 0];
% Characteristic function frequency range
+-
wmax=800;
% Number of discretizsation points
N=2^17; % Keep 2^k to use FFT i.s.o. DFT
% Average forward velocity in m/s
Vavg=20;
% Dimensions/System order
n=4;
% Number of Reference inputs
nu=2;
% Vehicle reference inputs (Required flight path)
Urefl=2*ones(1,22);
Uref2=linspace(2.2,4.4,22);
Uref=[Uref1;Uref2);
% Vehicle initial states
XO=[Uref(1,1) 0 -1.8 Vavg];
% Find simulation parameters based on above data (rule of thumb)
140
horizontime=0.11;
simtimestep=0.005;
manuvtimestep=horizontime/5;
X Get Vehicle Model
pcuavmodelextractl;
for t=[tO:simtimestep:totalsimtime-horizontime]
% Main simulation loop
% Next Horizon interval time vector
th=[t:simtimestep:t+horizontime-simtimestep];
Lt=length(th);
X Create Px and Mx time history over this horizon
Px=Pss;
% Make this more complicated later, if required
SYS = SS(A,Bu,C,D); % From pcuavmodelextractl function call
U=Uref(:,t/simtimestep+1:Lt+t/simtimestep);
Mx=LSIM(SYS,U,th-t,XO);
% Propagate and accum. Pc and Pcerr over horizon
X
Initialise prob of collision vector for this horizon
PcH=zeros(l,Lt);
X Initialize covariance vector for this horizon
Rho=zeros(1,Lt);
%Cor. Coef
Merr=zeros(1,Lt);
XMean error
Verr=zeros(1,Lt);
XVariance error
deltaPerr=zeros(1,Lt); % Change in Prob Col error
k=0;
for th=[t:simtimestep:t+horizontime-simtimestep]
% Begin Horizon Loop
k=k+1;
% Find density function of Gamma
[GamDist,GamVec,GamCharac]=finddist(Mx(k,:)',...
141
Px,Gam,wmax,N,0,0,zeros(1,n),1);
% Get GamDot into correct form for finddist
X No need when remnants used and added to tfm in finddist
% Find density function of GammaDot
EdeltaGamDotDist,deltaGamDotVec,deltaGamDotCharac] =f inddist (Mx (k,:)',...
Px,2*simtimestep*A.'*Gam,wmax,N,0,0,2*simtimestep*U(:,k).'*Bu.'*Gam,0);
X Calculation error estimates
% findrho finds quadratic form statistics
[Rho (k),stdA, stdB,meanA,meanB]=f indrho (Gam, simtimestep*GamD, Mx(k, :)'Px)
varA=(stdA)^2;
varB=(stdB)^2;
mslope=(-varB+varA+sqrt(varB^2-2*varA*varB+varA^2+...
4*Rho(k)^2*varA*varB))/2/Rho(k)/stdA/stdB;
if(k>1)
dP=PcH(k)-PcH(k-1);
ma=GamO; X A close approx, worst case is =0
Y.ma=0;
X Worst-case (these options exist to simplify)
deltam=1/mslope*dP/(1-dP)*(meanA-ma);
deltav=0; % Negligible effect ignored
else
deltam=0;
deltav=0;
end;
Merr(k)=deltam;
Verr(k)=deltav-2*Rho(k)*stdA*stdB;
% Accumulate Probability of Collision
if k>1
% Start Risk Accumulation
142
X NB: May REDUCE Calculations if Gam<O is not carried around!!
X Cut density function of Gamma into Df and Ds parts f=gl+g2
leq=( sqrt(varA+Verr(k)) )/stdA*GamO-Merr(k)
I=find(GamVecPrev>leq);
GamDistPrev=abs(GamDistPrev);
GamDistPrev=GamDistPrev/sum(GamDistPrev)*2*wmax;
gl=[GamDistPrev(:I(1)-1)
g2=[zeros(1,I(1)-1)
zeros(1,I(1))];
GamDistPrev(I(1):N)];
PosMinPc=sum(gl)/(2*wmax);
% Find Characteristic function of new Gamma
PropGamDistCut=g2/(1-PosMinPc);
PropGamVecNext=GamVec;
PropGamDistCut=[PropGamDistCut(N/2+1:N) PropGamDistCut(1:N/2)];
PropGamCharacCut=fft(PropGamDistCut);
X Plots for debugging
%figure;
%plot(GamVec,deltaGamDotDist);
% Convolve
PropGamCharacNext=PropGamCharacCut.*deltaGamDotCharacPrev;
PropGamDistNext=ifft(PropGamCharacNext);
X Flip, normalize and remove numerical IFFT imaginary errors
PropGamDistNext=[PropGamDistNext(N/2+1:N)
PropGamDistNext(1:N/2)];
PropGamDistNext=abs(PropGamDistNext);
PropGamDistNext=PropGamDistNext/sum(PropGamDistNext)*2*wmax;
X Plots for debugging
Xfigure;
%plot(PropGamVec,PropGamDistNext);
143
gi=[PropGamDistNext(i:I(1)-1) zeros(1,I(1))];
g2=[zeros(1,I(1)-1) PropGamDistNext(I(1):N)];
deltaPc=sum(g1)/(2*wmax);
PcH(k)=PcH(k-1)+deltaPc*(i-PcH(k-1));
X Pause for stepping and debugging
%pause;
else
leq=( sqrt(varA+Verr(k)) )/stdA*GamO-Merr(k)
I=find(GamVec>leq);
gI=[GamDist(i:I(1)-1) zeros(i,I(i))];
g2=[zeros(1,I(I)-i) GamDist(I(1):N)];
deltaPc=sum(gi) /(2*wmax);
PcH(k)=deltaPc;
% find risk for Prandini's lower bound
MinPc=deltaPc;
PosMinPc=O;
XJust initialize for subsequent if loop
PropGamDistNext=GamDist;
% End Risk Accumulation
end;
% Update Variables for next loop
GamVecPrev=GamVec;
GamDistPrev=GamDist;
deltaGamDotCharacPrev=deltaGamDotCharac;
% Find the Prandini's lower bound so far
if
PosMinPc>MinPc
MinPc=PosMinPc;
end;
144
if mod(k,1)==O
Xfigure;
disp(k);
%plot(GamVecPropGamDistNext,'b');
%hold on;
%plot(GamVec,GamDist,'r');
Xhold
off;
Xpause;
end;
% End Horizon loop
end;
if mod(t,0.5)==O
disp(t);
end;
end;
X End Main simulation loop
figure;
title{'Accumulated Prob. of Col. vs. sample'};
plot(PcH);
hold on;
plot(PcH-deltaPerr,'r');
plot(MinPc*ones(1,Lt),'k');
grid on;
zoom on;
legend('Upper Bound Pc','FOSM corrected Pc','Lower Bound Pc');
%title('Pc');
%figure;
145
%plot(Rho);
%title('Rho');
%figure;
%plot (Merr);
%title('Merr');
%figure;
%plot(Verr);
%title('Verr');
A.2
Sample Code for Calculation of Statistics of Two Dependent Quadratic Forms, "findrho.m":
% Thomas Jones
% Feb 2003
% Finding Covariance between symmetric
% quadratic forms: y=x'Ax and z=x'Bx
function [rho, stdA, stdB,meanA,meanB] =findrho(A,B,mx,sigx)
% Means
meanA=trace (A*sigx)+mx. '*A*mx;
meanB=trace (B*sigx)+mx. '*B*mx;
% Standard Deviations
stdA=sqrt (2*trace (A*sigx*A*sigx)+4*mx. '*A*sigx*A*mx);
stdB=sqrt (2*trace (B*sigx*B*sigx) +4*mx. '*B*sigx*B*mx);
X Correlation Coefficient
rho=(2*trace (A*sigx*B*sigx)+4*mx.'*A*sigx*B*mx)/stdA/stdB;
146
5Q3